Secure Transmission for Relay Wiretap Channels in the Presence of Spatially Random Eavesdroppers
\IEEEauthorblockN{Chenxi Liu\IEEEauthorrefmark{2
We propose a secure transmission scheme for a relay wiretap channel, where a source communicates with a destination via a decode-and-forward relay in the presence of spatially random-distributed eavesdroppers. We assume that the source is equipped with multiple antennas, whereas the relay, the destination, and the eavesdroppers are equipped with a single antenna each. In the proposed scheme, in addition to information signals, the source transmits artificial noise signals in order to confuse the eavesdroppers. With the target of maximizing the secrecy throughput of the relay wiretap channel, we derive a closed-form expression for the transmission outage probability and an easy-to-compute expression for the secrecy outage probability. Using these expressions, we determine the optimal power allocation factor and wiretap code rates that guarantee the maximum secrecy throughput, while satisfying a secrecy outage probability constraint. Furthermore, we examine the impact of source antenna number on the secrecy throughput, showing that adding extra transmit antennas at the source brings about a significant increase in the secrecy throughput.
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\begin{document}

\title{Secure Transmission for Relay Wiretap Channels in the Presence of
Spatially Random Eavesdroppers}

\author{\IEEEauthorblockN{Chenxi Liu\IEEEauthorrefmark{2}, Nan
Yang\IEEEauthorrefmark{3}, Jinhong Yuan\IEEEauthorrefmark{2}, and
Robert Malaney\IEEEauthorrefmark{2}}
\IEEEauthorblockA{\IEEEauthorrefmark{2}School of Electrical
Engineering and Telecommunications, The University of New South
Wales, Sydney, Australia\\\IEEEauthorrefmark{3}Research School of
Engineering, Australian National University, Canberra, Australia}
Email: chenxi.liu@student.unsw.edu.au, nan.yang@anu.edu.au,
j.yuan@unsw.edu.au, r.malaney@unsw.edu.au}

\maketitle

\begin{abstract}
We propose a secure transmission scheme for a relay wiretap channel,
where a source communicates with a destination via a
decode-and-forward relay in the presence of spatially random-distributed eavesdroppers. We assume that the source is equipped
with multiple antennas, whereas the relay, the destination, and the
eavesdroppers are equipped with a single antenna each. In the
proposed scheme, in addition to information signals, the source transmits artificial noise signals
in order to confuse the
eavesdroppers. With the target of maximizing the secrecy throughput
of the relay wiretap channel, we derive a closed-form expression for
the transmission outage probability and an easy-to-compute
expression for the secrecy outage probability. Using these
expressions, we determine the optimal power allocation factor and
wiretap code rates that guarantee the maximum secrecy throughput,
while satisfying a secrecy outage probability constraint.
Furthermore, we examine the impact of source antenna number on the secrecy throughput, showing that adding extra transmit
antennas at the source brings about a significant increase in the
secrecy throughput.
\end{abstract}

\IEEEpeerreviewmaketitle

\section{Introduction}

Wireless communications are inherently insecure, due to the
broadcast nature of the medium, which makes security a pivotal
design issue in the implementation and operation of current and
future wireless networks. Compared to the traditional key-based
cryptographic techniques that are applied to upper layers,
physical layer security can enhance the secrecy of wireless
communications without using secret keys and complex
encryption/decryption algorithms, and thus has been recognized as an
alternative for cryptographic techniques. The key idea of physical
layer security is to exploit the randomness of wireless channels to
offer secure data transmissions \cite{Hong,Yang_Mag}. In early
studies, e.g., \cite{wyner}, the principle of physical layer
security was established in a single-input single-output wiretap
channel. Subsequently, physical layer security in
multi-input multi-output (MIMO) communication systems has been
intensively investigated
\cite{khisti,wornell,chenxi,chenxi2,chenxi3,Zhou10,Zhang13,nan,nan2,nan3,nan4,shihao},
due to the benefits of MIMO techniques such as high data rate and
high link reliability.

Most recently, physical layer security in large-scale wireless
networks, such as mobile ad hoc and sensor networks, has been
receiving considerable attention
\cite{xiangyun_2,wang_he,gio_1,gio_2}. A key property of large-scale
wireless networks is that the node locations in the network were
spatially randomly distributed. As such, stochastic geometry and
random geometric graphs are used for modeling the locations of
spatially random-distributed nodes. In \cite{xiangyun_2}, the
throughput of large-scale decentralized wireless networks with
physical layer security constraints was investigated. Considering
the path loss as the sole factor influencing the received
signal-to-noise ratio (SNR) at the receiver, \cite{wang_he} examined
the secrecy rate in cellular networks. In \cite{gio_1} the secrecy rate achieved by linear precoding
was analyzed in the broadcast channel with spatially
random external eavesdroppers, and in \cite{gio_2} the secrecy rate achieved by linear precoding in cellular networks was analyzed.

%In \cite{gio_1} and
%\cite{gio_2}, the secrecy rate achieved by linear precoding was
%analyzed in the broadcast channel with spatially random external
%eavesdroppers and in the cellular networks, respectively.

We note
that \cite{xiangyun_2,wang_he,gio_1,gio_2} only considered
point-to-point transmissions. This leaves physical layer security
with cooperative relays in large-scale networks as an open problem.
Since the relay channel efficiently improves the coverage and
reliability in wireless networks \cite{Laneman,Bassily}, it is of
practical significance to investigate the secrecy performance of
such channels. Particularly important in this context would be extensions of previous work on relay wiretap channels which focused on scenarios where the location of an eavesdropper(s) is fixed and known at
the source (e.g.,
\cite{dong,xiaoming_1,xiaoming_2,yulong,hanzhu,chenxi4}).

%On the other hand, almost all the prior art on physical layer
%security in relay wiretap channels focused on the case where the
%location of the eavesdropper is fixed and known at the source
%\cite{dong,yulong,hanzhu,chenxi4}. In \cite{dong}, a distributed
%beamforming scheme was proposed to maximize the achievable secrecy
%rate of a relay wiretap channel, where the location of the
%eavesdropper was assumed to be fixed and known to the source.
%Considering relay selection, the secrecy outage probability was
%investigated in \cite{yulong} in a relay wiretap channel. In
%\cite{yulong}, the location of the eavesdropper was also assumed to
%be fixed and known at the source. Considering distributed multiple
%single-antenna two-way relays, a joint relay and jammer selection
%scheme was proposed in \cite{hanzhu} to maximize the sum secrecy
%rate. In \cite{hanzhu}, the location of the eavesdropper was assumed
%to be fixed and known at the source as well. In \cite{chenxi4}, a
%location-based secure transmission scheme was proposed to maximize
%the effective secrecy throughput of a relay wiretap channel, where
%the only information available on an eavesdropper is a noisy
%estimate of her location.

In this work we propose a secure transmission scheme for a relay
wiretap channel, where a source transmits to a destination via a
decode-and-forward (DF) relay in the presence of multiple spatially
random-distributed eavesdroppers. The source is
equipped with multiple antennas, while the relay, the destination,
and the eavesdropper are equipped with a single antenna each. In addition to information
signals, we assume that the source
transmit artificial noise signals in
order to confuse the eavesdroppers. Aiming at maximizing the secrecy throughput, while
satisfying a secrecy outage probability constraint, we determine
both the optimal power allocation factor between AN signals and
information signals and the optimal wiretap code rates.

Compared to current studies on physical layer security in relay
networks \cite{dong,xiaoming_1,xiaoming_2,yulong,hanzhu,chenxi4}, our contributions are threefold. First, we consider a more practical scenario where
the locations of the eavesdroppers are spatially randomly distributed
and not known at the source. Second, we propose a new secure
transmission scheme that maximizes the secrecy throughput in such a scenario. Third, we derive explicit expressions for the
transmission outage probability and the secrecy outage probability,
which enable us to analytically characterize the secrecy throughput
of the relay wiretap channel. We note that these expressions
are independent of realizations of the main channel and the
eavesdropper channel.

%Apart from the aforementioned contributions,
%we further examine the impact of the number of antenna at the source
%on the secrecy throughput, demonstrating that the secrecy throughput
%significantly increases with a higher number of antennas at the
%source.

The rest of the paper is organized as follows. Section
\ref{sec:system_model} describes the system model and presents the proposed scheme. In Section
\ref{sec:analysis}, we analyze and optimize the secrecy performance
achieved by the proposed scheme. Numerical results and related
discussions are presented in Section \ref{sec:numerical}. Finally,
Section \ref{sec:conclusion} draws conclusions.

\section{System Model and Proposed Scheme}\label{sec:system_model}

We consider a relay wiretap channel, as depicted in Fig.
\ref{system_model}, where a source (S) communicates with a
destination (D) with the aid of a relay (R) in the presence of
multiple spatially random eavesdroppers. We assume that the source
is equipped with $N$ antennas, while the destination, the relay, and
the eavesdroppers are equipped with a single antenna each. We also
assume that there is no direct link between the source and the
destination. We denote $d_{sr}$ and $d_{rd}$ as the distance between
the source and the relay and the distance between the relay and the
destination, respectively, and denote $\eta$ as the path loss
exponent. As in \cite{xiangyun_2,wang_he,gio_1,gio_2}, the eavesdroppers are modeled as a homogeneous Poisson
Point Process (PPP) $\Phi$ with density $\lambda$. This model is practical and representative for decentralized networks where each node is randomly distributed \cite{weber}. The source, the
relay, and the destination do not belong to $\Phi$.
%The locations of the source, the relay, and the destination are fixed.
\begin{figure}[!t]
\begin{center}{\includegraphics[width=0.9\columnwidth]{system_model.eps}}
\caption{Illustration of a relay wiretap channel with spatially
random eavesdroppers.}\label{system_model}
\end{center}
\end{figure}

\subsection{Transmission Scheme}

We now detail the proposed transmission scheme. The proposed scheme utilizes two time slots. In the first time
slot, the source transmits information signals together with AN
signals to the relay. In this time slot, the transmitted signals
from the source are overheard by the eavesdroppers. The role of the
AN signals is to confuse the eavesdroppers. In the second time slot, we assume that the source is silent. While it is clear that this will lead to a sub-optimal solution, such an assumption does allow for analytical tractability in a special case. Removing this assumption means more power can be added to the signal in the first time slot. Our other work \cite{chenxi5},  in which we consider multiple antennae at all nodes,  quantifies the advantage of adding source noise in the second time slot.
We also assume the relay transmits the received signals to the destination using
the DF protocol. In this time slot, the broadcast signals are also
overheard by the eavesdroppers. We assume that all the channels are
subject to independent and identically distributed Rayleigh fading.
We also assume a quasi-static block fading environment in which all
the channel coefficients remain the same within one time slot. We
denote $\mathbf{h}_{sr}$ as the $1\times N$ channel vector from the
source to the relay and denote $h_{rd}$ as the channel coefficient
from the relay to the destination.

We now express the transmitted and received signals in two time
slots separately. In the first time slot, the transmitted signal at
the source is given by
\begin{align}\label{x_s}
\mathbf{x}_{\text{S}} = \mathbf{W}\mathbf{t},
\end{align}
where $\mathbf{W}$ denotes the $N\times N$ beamforming matrix at the
source and $\mathbf{t}$ denotes the combination of the information
signal and the AN signal. To perform a such transmission, we first
design $\mathbf{W}$ as
\begin{align}\label{AN_beamforming}
\mathbf{W}=\begin{bmatrix}
\mathbf{w}_{\text{S}}&\mathbf{W}_{\text{AN}}
\end{bmatrix},
\end{align}
where $\mathbf{w}_{\text{S}}$ is used to transmit the information
signal and $\mathbf{W}_{\text{AN}}$ is used to transmit the AN
signal. The aim of $\mathbf{W}$ is to degrade the quality of the
received signals at the eavesdroppers. By transmitting AN signals
into the null space of $\mathbf{h}_{sr}$ through $\mathbf{W}$, the
source ensures that the quality of the received signals at the relay
is free from AN interference. In designing $\mathbf{W}$, we
choose $\mathbf{w}_{\text{S}}$ as the eigenvector corresponding to
the largest eigenvalue of $\mathbf{h}_{sr}^H\mathbf{h}_{sr}$. We
then choose $\mathbf{W}_{\text{AN}}$ as the remaining $N-1$
eigenvectors of $\mathbf{h}_{sr}^H\mathbf{h}_{sr}$. Such a design
ensures that $\mathbf{W}$ is a unitary matrix.
%We also note that the knowledge of $\mathbf{h}_{sr}$ is required to be available at the source in order to design such $\mathbf{W}$.
We then design $\mathbf{t}$ as
\begin{align}\label{t_s}
\mathbf{t}=\begin{bmatrix} t_{\text{S}}\\ \mathbf{t}_{\text{AN}}
\end{bmatrix},
\end{align}
where $t_{\text{S}}$ denotes the information signal and
$\mathbf{t}_{\text{AN}}$ is an $\left(N-1\right)\times 1$ vector of
the AN signal. We define $\beta$, $0<\beta\leq 1$, as the fraction
of the power allocated to the information signal. As such, we obtain
$\mathbb{E}\left[|t_{\text{S}}|^2\right]=\beta$ and
$\mathbb{E}\left[\mathbf{t}_{\text{AN}}\mathbf{t}_{\text{AN}}^H\right]=\frac{1-\beta}{N-1}\mathbf{I}_{N-1}$,
where $\mathbb{E}\left[\cdot\right]$ is expectation and
$\mathbf{I}_{m}$ is the $m\times{m}$ identity matrix. Moreover, we
confirm that
$\mathbb{E}\left[\mathbf{x}_{\text{S}}\mathbf{x}_{\text{S}}^H\right]=\mathbf{I}_N$.
Based on \eqref{x_s}, \eqref{AN_beamforming}, and \eqref{t_s}, the
received signal at the relay in the first time slot is expressed as
\begin{align}\label{signal_relay}
y_r =
\sqrt{P_sd_{sr}^{-\eta}}\mathbf{h}_{sr}\mathbf{w}_{\text{S}}t_{\text{S}}
+ n_r,
\end{align}
where $P_s$ denotes the transmit power at the source, and $n_r$ denotes the thermal noise at the relay, which is assumed to be a zero
mean complex Gaussian random variable with variance $\sigma_r^2$,
i.e., $n_r\sim\mathcal{CN}\left(0,\sigma_r^2\right)$.

We next express the received signal at a typical eavesdropper
located at $i$, $i\in\Phi$, in the first time slot as
\begin{align}
\label{signal_i} y_i^{(1)} =
\sqrt{P_sd_{si}^{-\eta}}\mathbf{h}_{si}\mathbf{w}_{\text{S}}t_{\text{S}}
+
\sqrt{P_sd_{si}^{-\eta}}\mathbf{h}_{si}\mathbf{W}_{\text{AN}}\mathbf{t}_{\text{AN}}
+n_{i1},
\end{align}
where $\mathbf{h}_{si}$ denotes the $1\times N$ channel vector from
the source to the typical eavesdropper located at $i$, $d_{si}$
denotes the distance between the source and the typical eavesdropper
located at $i$, and $n_{i1}$ denotes the thermal noise at the
typical eavesdropper located at $i$, which is assumed to be a zero mean complex
Gaussian random variable with variance $\sigma_{i1}^2$, i.e.,
$n_{i1}\sim\mathcal{CN}\left(0,\sigma_{i1}^2\right)$.

In the second time slot, the relay adopts the DF protocol to
forward signals to the destination.
Specifically, the relay first decodes the transmitted signals from
the source and then broadcasts them after re-encoding. Therefore, we
express the received signal at the destination as
\begin{align}\label{signal_d}
y_d=\sqrt{P_rd_{rd}^{-\eta}}h_{rd}x_{\text{R}}+n_d,
\end{align}
where $P_r$ denotes the transmit power at the relay, $h_{rd}$ denotes
the channel coefficient from the relay to the destination, $x_{\text{R}}$ denotes the transmitted signal of the relay
with $\mathbb{E}\left[\|x_{\text{R}}\|^2\right]=1$, and $n_d$
denotes the thermal noise at the destination, which is assumed to be a zero mean
complex random variable with variance $\sigma_d^2$, i.e.,
$n_d\sim\mathcal{CN}\left(0,\sigma_d^2\right)$.

We next express the received signal in the second time slot at the typical eavesdropper
located at $i$ is expressed as
\begin{align}\label{signal_i2}
y_i^{(2)}=\sqrt{P_rd_{ri}^{-\eta}}h_{ri}x_{\text{R}}+n_{i2},
\end{align}
where $h_{ri}$ denotes the channel coefficient from the relay to the
typical eavesdropper located at $i$, $d_{ri}$ denotes the distance
between the relay and the typical eavesdropper located at $i$, and
$n_{i2}$ denotes the thermal noise at the the typical eavesdropper
located at $i$, which is assumed to be a zero mean complex random variable with
variance $\sigma_{i2}^2$, i.e.,
$n_{i2}\sim\mathcal{CN}\left(0,\sigma_{i2}^2\right)$.
%Without loss of generality, we assume $\sigma_r^2=\sigma_d^2=\sigma_{i1}^2=\sigma_{i2}^2=1$.

\subsection{Received SNRs}

We express the
instantaneous SNR at the destination as
\begin{align}
\label{snr_d} \Gamma_D = \min\left\{\gamma_{sr},\gamma_{rd}\right\},
\end{align}
where
$\gamma_{sr}=\frac{\beta{}P_s}{d_{sr}^{\eta}\sigma_r^{2}}\|\mathbf{h}_{sr}\|^2$
and
$\gamma_{rd}=\frac{P_r}{d_{rd}^{\eta}\sigma_d^{2}}\|{h}_{rd}\|^2$.
%\begin{align}\label{snr_sr}
%\gamma_{sr}=\frac{\beta{}P_s}{d_{sr}^{\eta}\sigma_r^{2}}\|\mathbf{h}_{sr}\|^2
%\end{align}
%and
%\begin{align}\label{snr_rd}
%\gamma_{rd}=\frac{P_r}{d_{rd}^{\eta}\sigma_d^{2}}\|{h}_{rd}\|^2.
%\end{align}
For the instantaneous SNR at the eavesdroppers, we assume that the
eavesdroppers are non-colluding, which means that each
eavesdropper decodes her own received signals from the source and
the relay, without cooperating with other eavesdroppers. Thus, we express the instantaneous SNR at the
eavesdroppers as\footnote{We assume that the source and the relay use different wiretap codes with
different codebooks. As such, the transmitted signals from the source and relay cannot be jointly processed by any  eavesdropper (nor by any combination of eavesdroppers).}
\begin{align}\label{snr_e}
\Gamma_E=\max_{i\in\Phi}\left\{\max\left\{\gamma_{si},\gamma_{ri}\right\}\right\},
\end{align}
where
\begin{align}\label{snr_si}
\gamma_{si}=\frac{\beta{}P_sd_{si}^{-\eta}\|\mathbf{h}_{si}\mathbf{w}_{\text{S}}\|^2}
{\frac{1-\beta}{N-1}P_sd_{si}^{-\eta}\mathbf{h}_{si}\mathbf{W}_{\text{AN}}\mathbf{W}_{\text{AN}}^H\mathbf{h}_{si}^H+\sigma_{i1}^2}.
\end{align}
and
$\gamma_{ri}=\frac{P_r}{d_{ri}^{\eta}\sigma_{i2}^2}\|h_{ri}\|^2$.
%\begin{align}\label{snr_ri}
%\gamma_{ri}=\frac{P_r}{d_{ri}^{\eta}\sigma_{i2}^2}\|h_{ri}\|^2.
%\end{align}

\subsection{Problem Formulation}

In order to evaluate and optimize the secrecy performance achieved
by our proposed scheme, we apply the
performance metric proposed in \cite{Zhou10}, which is given by
\begin{align}\label{throughput}
T_s=\frac{1}{2}\left(R_b-R_e\right)\left(1-P_{to}\right),
\end{align}
where the factor $1/2$ is due to the two time slots used in the transmission, $\left(R_b,R_e\right)$ denotes a parameter pair of the wiretap
code used by the source, $R_b$ denotes the transmission rate of the
wiretap code, $R_e$ denotes the cost of preventing the transmitted
wiretap code from eavesdropping, and $P_{to}$ denotes the
transmission outage probability ($P_{to}$ is defined as the probability
that the instantaneous SNR at the destination is less than
$\tau_b = 2^{R_b}-1$). Mathematically, $P_{to}$ is formulated as
\begin{align}\label{p_to}
P_{to}={\Pr}\left(\Gamma_{D}\leq\tau_b\right).
\end{align}
Henceforth, we refer to $T_s$ as the secrecy throughput.

The goal of this work is to maximize the secrecy throughput of the relay
wiretap channel with spatially random eavesdroppers under secrecy
constraints. To achieve this goal, we formulate the design problem
as
\begin{align}\label{prob_form}
\max_{R_b,R_e,\beta}& ~~T_s,\notag\\
s.t.~~P_{so}\leq \varphi, 0\leq R_e&\leq R_b, 0<\beta\leq1,
\end{align}
where $P_{so}$ denotes the secrecy outage probability. $P_{so}$ is
defined as the probability that $\Gamma_{E}$ is larger than $\tau_e
= 2^{R_e}-1$. Mathematically, $P_{so}$ is formulated as
\begin{align}\label{p_sec}
P_{so}={\Pr}\left(\Gamma_{E}>\tau_e\right).
\end{align}
%We note that in the design problem, the secrecy outage probability
%is required to be lower than or equal to a secrecy outage
%probability constraint $\varphi$.

\section{Analysis and Optimization of Secrecy Performance}\label{sec:analysis}

In this section, we first analyze the secrecy performance by
deriving explicit expressions for the transmission probability and
the secrecy outage probability, respectively. Based on these
results we determine the optimal parameters, e.g., $R_b$, $R_e$,
and $\beta$, that achieve the optimal secrecy performance of the
relay wiretap channel. Notably, the determined optimal secrecy
performance is independent of realizations of the main channel and
the eavesdroppers' channels.

%Moreover, we examine the asymptotic transmission outage probability and the asymptotic secrecy outage probability when the number of antenna at the source is relatively large, i.e. $N\to\infty$.

\subsection{Transmission Outage Probability}

In this subsection we focus on the transmission outage probability,
$P_{to}$. We first obtain the cumulative distribution functions
(CDFs) of $\gamma_{sr}$ and $\gamma_{rd}$ as
\begin{align}\label{cdf_1}
F_{\gamma_{sr}}\left(\gamma\right)=1-\exp\left(-\frac{\gamma}{\beta\overline{\gamma}_{sr}}\right)
\sum_{n=0}^{N-1}\frac{1}{n!}\left(\frac{\gamma}{\beta\overline{\gamma}_{sr}}\right)^n
\end{align}
with $\overline{\gamma}_{sr}=P_sd_{sr}^{-\eta}\sigma_r^{-2}$, and
\begin{align}\label{cdf_2}
F_{\gamma_{rd}}\left(\gamma\right)=1-\exp\left(-\frac{\gamma}{\overline{\gamma}_{rd}}\right)
\end{align}
with $\overline{\gamma}_{rd}=P_rd_{rd}^{-\eta}\sigma_d^{-2}$.
Based on \eqref{snr_d}, \eqref{cdf_1}, and \eqref{cdf_2}, we
re-express the transmission outage probability in \eqref{p_to} as
\begin{align}\label{p_tr_2}
P_{to}=&{\Pr}\left(\min\left\{\gamma_{sr},\gamma_{rd}\right\}\leq\tau_b\right)\notag\\
=&1-\left(1-F_{\gamma_{sr}}\left(\tau_b\right)\right)\left(1-F_{\gamma_{rd}}\left(\tau_b\right)\right)\notag\\
=&1-\exp\left(-\left(\frac{1}{\beta\overline{\gamma}_{sr}}+\frac{1}{\overline{\gamma}_{rd}}\right)\tau_b\right)
\sum_{n=0}^{N-1}\frac{1}{n!}\left(\frac{\tau_b}{\beta\overline{\gamma}_{sr}}\right)^n.
\end{align}

\subsection{Secrecy Outage Probability}

We now focus on the secrecy outage probability, $P_{so}$. To derive
$P_{so}$, we first express the CDFs of $\gamma_{si}$ and
$\gamma_{ri}$ as
\begin{align}\label{cdf_3}
F_{\gamma_{si}}\left(\gamma\right)=1-\left(1+\frac{\left(1-\beta\right)\gamma}
{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\exp\left(-\frac{\gamma}{\beta\overline{\gamma}_{si}}\right)
\end{align}
with $\overline{\gamma}_{si}=P_sd_{si}^{-\eta}\sigma_{i1}^{-2}$, and
\begin{align}\label{cdf_4}
F_{\gamma_{ri}}\left(\gamma\right)=1-\exp\left(-\frac{\gamma}{\overline{\gamma}_{ri}}\right)
\end{align}
with $\overline{\gamma}_{ri} = P_rd_{ri}^{-\eta}\sigma_{i2}^{-2}$.
Based on \eqref{snr_e}, \eqref{cdf_3} and \eqref{cdf_4}, we
present the secrecy outage probability in the following theorem.
\begin{theorem}\label{t1}
The secrecy outage of the relay wiretap channel is derived as
\begin{align}\label{t1_result}
P_{so}=1-\exp\left(-2\lambda\left(\mathcal{J}_1 + \mathcal{J}_2 - \mathcal{J}_3\right)\right),
\end{align}
where
\begin{align}\label{t1_result_2}
\mathcal{J}_{1}=&\frac{\pi}{\eta}\left(\frac{\beta
P_s}{\tau_e\sigma_{i1}^2}\right)^{\frac{2}{\eta}}
\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\Gamma\left(\frac{2}{\eta}\right),
\end{align}
%\begin{align}\label{t1_result_3}
%&\mathcal{J}_2\notag\\
%=&\int_{0}^{\infty}\int_{0}^{\pi}\!\!\!d_{si}\exp\!\left(\!\!-\frac{\tau_e\sigma_{i2}^2}{P_r}\left(d_{sr}^2\!+\!d_{si}^2\!-\!2d_{sr}d_{si}\cos\theta\right)^{\frac{\eta}{2}}\!\!\right)dd_{si}d\theta,
%\end{align}
%\begin{align}\label{t1_result_4}
%&\hspace{0.35cm}\mathcal{J}_3\notag\\
%&=\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\int_{0}^{\infty}\int_{0}^{\pi}d_{si}\exp\left(-\frac{\tau_e\sigma_{i1}^2}{\beta P_s}d_{si}^{\eta}\right)\notag\\
%&\times\exp\!\left(\!-\frac{\tau_e\sigma_{i2}^2}{P_r}\left(d_{sr}^2\!+\!d_{si}^2\!-\!2d_{sr}d_{si}\cos\theta\right)^{\frac{\eta}{2}}\!\right)\!dd_{si}d\theta,
%\end{align}
\begin{align}\label{t1_result_3}
\mathcal{J}_{2}=\int_{0}^{\infty}\int_{0}^{\pi}d_{si}\exp\left(-\psi\left(\theta\right)\right)dd_{si}d\theta,
\end{align}
\begin{align}\label{t1_result_4}
\mathcal{J}_{3}=&\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\notag\\
&\times\int_{0}^{\infty}\int_{0}^{\pi}d_{si}\exp\left(-\frac{\tau_e\sigma_{i1}^2}{\beta{}P_s}d_{si}^{\eta}\right)
\exp\left(-\psi\left(\theta\right)\right)dd_{si}d\theta,
\end{align}
and
$\psi\left(\theta\right)=\frac{\tau_e\sigma_{i2}^2}{P_r}\left(d_{sr}^2+d_{si}^2-2d_{sr}d_{si}\cos\theta\right)^{\frac{\eta}{2}}$.
\begin{IEEEproof}
See Appendix \ref{App_t1}.
\end{IEEEproof}
\end{theorem}

We find that \eqref{t1_result} provides an easy-to-compute
expression for the secrecy outage probability. Despite that
$\mathcal{J}_2$ and $\mathcal{J}_3$ for general $\eta$ can not be
obtained in closed-form, they can be easily
evaluated since only a double integral is
involved in $\mathcal{J}_2$ and $\mathcal{J}_3$.

We next present simplified closed-form expressions for the special
case where $\eta=2$. For this special case, we first simplify
$\mathcal{J}_2$, which yields a closed-form expression given by
\eqref{J_2_fur} (next page),
\begin{figure*}[ht]
\begin{align}\label{J_2_fur}
\mathcal{J}_2 = & \exp\left(-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{sr}^2\right)\int_0^{\infty}d_{si}\exp\left(-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{si}^2\right)
\int_{0}^{\pi}\exp\left(\frac{2\tau_e\sigma_{i2}^2}{P_r}d_{sr}d_{si}\cos\theta\right)d\theta dd_{si}\notag\\
\overset{(a)}{=}&\pi\exp\left(-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{sr}^2\right)\int_0^{\infty}d_{si}\exp\left(-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{si}^2\right)
I_{0}\left(\frac{2\tau_e\sigma_{i2}^2}{P_r}d_{sr}d_{si}\right)dd_{si}\notag\\
\overset{(b)}{=}&\pi\exp\left(-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{sr}^2\right)\int_{0}^{\infty}d_{si}\exp\left(-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{si}^2\right)
\sum_{k=0}^{\infty}\frac{1}{\left(k!\right)^2}\left(\frac{\tau_e\sigma_{i2}^2}{P_r}d_{sr}d_{si}\right)^{2k}dd_{si}\notag\\
\overset{(c)}{=}&\frac{\pi P_r}{2\tau_e\sigma_{i2}^2}\exp\left(-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{sr}^2\right)
\sum_{k=0}^{\infty}\frac{1}{\left(k!\right)^2}\left(\frac{\tau_e\sigma_{i2}^2}{P_r}d_{sr}^2\right)^k\Gamma\left(k+1\right)\notag\\
\overset{(d)}{=}&\frac{\pi P_r}{2\tau_e\sigma_{i2}^2}.
\end{align}
\hrule
\end{figure*}
where $(a)$ follows by applying the expression of the zero-order
modified Bessel function $I_0(z)$, $(b)$
follows by applying the series representation of $I_0(z)$, $(c)$ follows by  using $t=\frac{\tau_e\sigma_{i2}^2}{P_r}d_{si}^{2}$ and noting that
\begin{align}
\int_{0}^{\infty}t^{k}\exp\left(-t\right)dt=\Gamma\left(k+1\right),
\end{align}
%applying the series representation of $I_0(z)$, $(c)$ follows by applying the expression of gamma
%function,
and $(d)$ follows by applying the series expression representation
of the exponential function. Similarly,
for the special case where $\eta=2$, we simplify $\mathcal{J}_3$,
which yields a closed-form expression given by \eqref{J_3_fur} (next page). The simplified expressions in \eqref{J_2_fur} and
\eqref{J_3_fur} offer us a computationally efficient way to calculate the secrecy outage probability for the special case where $\eta=2$.
\begin{figure*}[ht]
\begin{align}\label{J_3_fur}
\mathcal{J}_3 = & \left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\!\!\exp\!\left(\!-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{sr}^2\!\right)\!
\int_{0}^{\infty}\!d_{si}\exp\!\left(\!-\left(\frac{\tau_e\sigma_{i1}^2}{\beta P_s}+\frac{\tau_e\sigma_{i2}^2}{P_r}\right)d_{si}^2\!\right)\!
\int_{0}^{\pi}\exp\left(\frac{2\tau_e\sigma_{i2}^2}{P_r}d_{sr}d_{si}\cos\theta\right)d\theta d d_{si}\notag\\
=&\frac{\pi\beta P_s P_r}{2\tau_e\left(\beta P_s\sigma_{i2}^2 + P_r\sigma_{i1}^2\right)}
\exp\left(-\frac{\tau_e\sigma_{i1}^2\sigma_{i2}^2}{\beta P_s\sigma_{i2}^2+P_r\sigma_{i1}^2}d_{sr}^2\right)
\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}.
\end{align}
\hrule
\end{figure*}

\subsection{Throughput Optimization}

In this subsection we determine the optimal parameters that
maximize the secrecy throughput, $T_s$, for general $\eta$. Specifically, we first determine the optimal wiretap code rates
pair, $\left(R_b^{\ast},R_e^{\ast}\right)$, that maximizes the
secrecy throughput for a given power allocation factor $\beta$. We
then determine the joint optimal power allocation factor and wiretap
code rates,
$\left(\beta^{\ast\circ},R_b^{\ast\circ},R_e^{\ast\circ}\right)$,
that maximizes $T_s$.

\subsubsection{Optimal wiretap code rates pair for a given $\beta$}

The optimal wiretap code rates pair,
$\left(R_b^{\ast},R_e^{\ast}\right)$, that maximizes $T_s$ for a given $\beta$ is determined as
\begin{align}\label{prob_form_2}
\left(R_b^{\ast},R_e^{\ast}\right) = \argmax~ T_s,\notag\\
s.t.~~ P_{so}\leq\varphi, 0\leq R_e\leq R_b.
\end{align}
Taking the first-order derivative of $P_{so}$ with respect to $R_e$,
we confirm that $\partial P_{so}/\partial R_e<0$, which states
that $P_{so}$ monotonically decreases as $R_e$ increases. As such,
the value of $R_e^{\ast}$ satisfying \eqref{prob_form_2} is the
value of $R_e^{\ast}$ that satisfies the secrecy outage probability
constraint, i.e., $P_{so}\left(R_e^{\ast}\right)=\varphi$. We
then confirm that $\partial T_s/\partial R_b$ is first positive then
negative as $R_b$ increases. This demonstrates that the value of
$R_b^{\ast}$ satisfying \eqref{prob_form_2} is unique. Although a
closed-form solution for $\left(R_b^{\ast},R_e^{\ast}\right)$ is
mathematically intractable, we are able to determine the values of
$\left(R_b^{\ast},R_e^{\ast}\right)$ numerically. Accordingly, the
maximal secrecy throughput based on the values of
$\left(R_b^{\ast},R_e^{\ast}\right)$ for a given $\beta$ is defined
as $T_s^{\ast}$.

\subsubsection{Joint optimization of $\beta$, $R_b$, and $R_e$}

The joint optimal power allocation factor and wiretap code rates
which maximizes $T_s$ in \eqref{throughput},
$\left(\beta^{\ast\circ},R_b^{\ast\circ},R_e^{\ast\circ}\right)$, is
determined as
\begin{align}\label{prob_form_3}
\left(\beta^{\ast\circ},R_b^{\ast\circ},R_e^{\ast\circ}\right)& = \argmax~ T_s,\notag\\
s.t.~~ P_{so}\leq\varphi, ~0\leq R_e&\leq R_b, ~0<\beta\leq1.
\end{align}
Using \eqref{p_tr_2} and \eqref{t1_result}, we are able to solve
\eqref{prob_form_3} numerically. Specifically, we first determine
the value of $\left(R_b^{\ast},R_e^{\ast}\right)$ using
\eqref{prob_form_2} for each value of $\beta$. This leads to the
secrecy throughput with $\left(R_b^{\ast},R_e^{\ast}\right)$,
denoted by
$T_s^{\ast}=\frac{1}{2}\left(R_b^{\ast}-R_e^{\ast}\right)\left(1-P_{to}\right)$.
We then determine the value of $\beta$ that maximizes $T_s^{\ast}$,
denoted by $\beta^{\ast\circ}$. Accordingly, the value of
$\left(R_b^{\ast},R_e^{\ast}\right)$ associated with
$\beta^{\ast\circ}$ is determined as
$\left(R_b^{\ast\circ},R_e^{\ast\circ}\right)$. Finally, the maximal
secrecy throughput based on the values of $\beta^{\ast\circ}$,
$R_b^{\ast\circ}$, and $R_e^{\ast\circ}$ is defined as
$T_s^{\ast\circ}$.

\section{Numerical Results}\label{sec:numerical}

In this section we present numerical results to validate our
analysis of the outage probabilities and examine the benefits of the
proposed scheme. For illustrative purpose, throughout this section we concentrate on the
practical example of a highly shadowed urban area with $\eta = 4$. In addition, we adopt $\lambda d_{sr}^2 = \lambda d_{rd}^2 =
1$.

\begin{figure}[t!]
\begin{center}{\includegraphics[height=2.3in,width=3.0in]{transmission_outage_1.eps}}
\caption{$P_{to}$ versus $\tau_b$ for different values of
$\overline{\gamma}_b$ with $\eta=4$, $N=4$, and $\beta =
0.5$.}\label{fig_side_a}
\end{center}
\end{figure}

We first verify the accuracy of the transmission outage probability
and the secrecy outage probability using Monte Carlo simulations. In
Fig. \ref{fig_side_a}, we plot $P_{to}$ versus $\tau_b$ for
different values of $\overline{\gamma}_b$ with $N=4$ and $\beta=0.5$. In this
figure, we consider
$\overline{\gamma}_{sr}=\overline{\gamma}_{rd}=\overline{\gamma}_b$.
We first see that the analytical curves, generated from
\eqref{p_tr_2}, precisely match the simulation points, which
demonstrates the correctness of our expression for $P_{to}$ in
\eqref{p_tr_2}. Second, we see that $P_{to}$ increases monotonically
as $\tau_b$ increases for a given $\overline{\gamma}_b$, which
implies that the transmission outage probability increases when the
transmission rate of the wiretap code increases. We further see that
$P_{to}$ decreases as $\overline{\gamma}_b$ increases for a given
$\tau_b$. This reveals that the transmission outage probability
reduces when the source and the relay use more power to transmit
under a fixed $\tau_b$.

\begin{figure}[t!]
\begin{center}{\includegraphics[height=2.3in,width=3.0in]{secrecy_outage_1.eps}}
\caption{$P_{so}$ versus $\tau_e$ for different values of
$\overline{\gamma}_e$ with $\eta=4$, $N=4$, and $\beta =
0.5$.}\label{fig_side_b}
\end{center}
\end{figure}

In Fig. \ref{fig_side_b}, we plot $P_{so}$ versus $\tau_e$ for
different values of $\overline{\gamma}_e$ with $N=4$ and $\beta = 0.5$. In
this figure we consider
$\overline{\gamma}_{sr}{\sigma_r^2}/{\sigma_{i1}^2}=\overline{\gamma}_{rd}{\sigma_d^2}/{\sigma_{i2}^2}=\overline{\gamma}_e$.
We see an excellent match between the analytical curves
generated from \eqref{t1_result} and the simulation points,
demonstrating the correctness of our expression for $P_{so}$ in
\eqref{t1_result}. We then see that $P_{so}$ decreases monotonically
as $\tau_e$ increases for a given $\overline{\gamma}_e$, which
shows that the secrecy outage probability decreases when the
redundancy rate of the wiretap code increases. We further observe
that $P_{so}$ increases as $\overline{\gamma}_e$ increases. This is
due to the fact the eavesdroppers receive signals from {\it both} the
source and the relay. It follows that increasing the transmit power
at the source and the relay leads to an improved received SNR at the
eavesdroppers.

\begin{figure}[t!]
\begin{center}{\includegraphics[height=2.3in,width=3.0in]{secrecy_throughput_Rb_2.eps}}
\caption{$T_s$ versus $R_b$ for different values of $N$ with
$\eta=4$, $\beta = 0.5$, $\varphi = 0.4$, $\overline{\gamma}_b = 20$
dB, and $\overline{\gamma}_b/\overline{\gamma}_e =
20$.}\label{fig_side_c}
\end{center}
\end{figure}

We now examine the impact of the system parameters $R_b$ and $\beta$ on
the secrecy throughput. In Fig. \ref{fig_side_c}, we plot $T_s$
versus $R_b$ for different values of $N$ with a fixed $\beta$ and the optimal $R_e$. We first observe that there exists a
unique $R_b^{\ast}$ that maximizes $T_s$ for a given $\beta$. We
also observe that the maximal $T_s$ for a given $\beta$, i.e.,
$T_s^{\ast}$, increases as $N$ increases. This shows that adding
extra transmit antennas at the source significantly enhances the
secrecy performance of the relay wiretap channel.

\begin{figure}[t!]
\begin{center}{\includegraphics[height=2.3in,width=3.0in]{secrecy_throughput_beta_2.eps}}
\caption{$T_s^{\ast}$ versus $\beta$ for different values of $N$
with $\eta=4$, $\varphi = 0.4$, $\overline{\gamma}_b = 20$ dB, and
$\overline{\gamma}_b/\overline{\gamma}_e = 20$.}\label{fig_side_d}
\end{center}
\end{figure}

In Fig. \ref{fig_side_d}, we plot $T_s^{\ast}$ versus $\beta$ for
different values of $N$. For each point of $T_s^{\ast}$, we choose
the $\left(R_b^{\ast},R_e^{\ast}\right)$ pair that maximizes $T_s$ for the
corresponding $\beta$. We first observe that there exists a unique
$\beta^{\ast\circ}$ that maximizes $T_s^{\ast}$. We then observe
that the value of $\beta^{\ast\circ}$ that maximizes $T_s^{\ast}$ is
around $0.9$, which demonstrates that the secrecy throughput is
improved if approximately 10\% of the total transmit power at the
source is allocated to AN signals. We also observe that the maximal
$T_s^{\ast}$, i.e., $T_s^{\ast\circ}$, increases as $N$ increases.
Furthermore, we observe that the value of $\beta^{\ast\circ}$
slightly decreases as $N$ increases, which shows that in order to maintain the optimal secrecy throughput, the power allocated to AN must increase as the source antenna number increases.

%In Fig. \ref{fig_side_c}, we examine the impact of the wiretap
%transmission rate $R_b$ on the secrecy throughput for a given power
%allocation factor $\beta$. In this figure, we plot $T_s$ versus
%$R_b$ for different values of $N$ when $\beta$ is fixed and $R_e$ is
%optimized to satisfy the secrecy outage probability constraint. We
%first observe that there exists a unique $R_b^{\ast}$ that maximizes
%$T_s$ for a given $\beta$. We also observe that the maximal $T_s$
%for a given $\beta$, denoted by $T_s^{\ast}$, increases as $N$
%increases, which indicates that adding extra transmit antenna
%significantly increases the secrecy performance of the relay wiretap
%channel.

%In Fig. \ref{fig_side_d}, we examine the impact of the power
%allocation factor $\beta$ on the secrecy throughput. In this figure,
%we plot $T_s^{\ast}$ versus $\beta$ for different values of $N$. For
%each point of $T_s^{\ast}$, $\left(R_b^{\ast},R_e^{\ast}\right)$
%that maximizes $T_s$ is chosen for the corresponding $\beta$. As
%expected, we first observe that there exists a unique
%$\beta^{\ast\circ}$ that maximizes $T_s^{\ast}$. We then observe
%that the $\beta^{\ast\circ}$ that maximizes $T_s^{\ast}$ is around
%$0.9$, which indicates that a small fraction of the transmit power
%of the source devoted to AN results in a significant increase in the
%secrecy throughput. We also observe that the maximal $T_s^{\ast}$,
%denoted by $T_s^{\ast\circ}$, increases as $N$ increases. Moveover,
%we observe that the value of $\beta^{\ast\circ}$ that maximizes
%$T_s^{\ast}$ decreases as $N$ increases, which indicates that more
%power needs to be devoted to AN when the source is equipped with
%more antenna in order to achieve the optimal secrecy performance of
%the relay wiretap channel.

\section{Conclusion}\label{sec:conclusion}

In this work we proposed a secure transmission scheme for a
relay wiretap channel, in which an $N$-antenna source transmits both
information signals and AN signals in the presence of multiple
spatially random single-antenna eavesdroppers. Conditioned on the use of a decode-and-forward protocol at the relay, we determined the optimal
parameters
$\left(\beta^{\ast\circ},R_b^{\ast\circ},R_e^{\ast\circ}\right)$
that maximizes the secrecy throughput,
based on our derived expressions for the transmission outage
probability and the secrecy outage probability. In addition, we
examined the impact of $N$ on the secrecy throughput, showing how
the maximal secrecy throughput increases with $N$. The work reported here provides valuable insights into the design of new physical layer security schemes in which the locations of the eavesdroppers are randomly distributed and not known at the source.

%to a destination with the aid of a DF relay in the presence of
%multiple spatially random eavesdroppers. We assumed that the source
%is equipped with multiple antennas, while the destination, the
%relay, and the eavesdropper are all equipped with a single antenna
%each. In our scheme, the source transmits AN to confuse the
%eavesdropper. Using the secrecy throughput as the performance
%metric, we determined the optimal parameters
%$\left(\beta^{\ast\circ},R_b^{\ast\circ},R_e^{\ast\circ}\right)$
%that maximizes the secrecy throughput of the relay wiretap channel,
%where $\beta^{\ast\circ}$, $R_b^{\ast\circ}$, and $R_e^{\ast\circ}$
%denote the optimal power allocation factor, the optimal wiretap code
%transmission rate, and the optimal wiretap code redundancy rate,
%respectively. In addition, we examined the impact of the number of
%antenna at the source on the secrecy throughput, demonstrating how
%the maximal secrecy throughput increases as the number of antenna at
%the source increases.

\section*{Acknowledgements}
The work of J. Yuan and R. Malaney was funded by the Australian Research Council Discovery Project DP120102607. The work of N. Yang was funded by the Australian Research Council Discovery Project DP150103905, and the Ian Potter Foundation's travel grant 20160251.

\begin{appendices}

\section{Proof of Theorem \ref{t1}}\label{App_t1}

According to \eqref{snr_e}, \eqref{cdf_3} and \eqref{cdf_4}, we
re-express \eqref{p_sec} as
\begin{align}\label{p_sec_2}
P_{so}&=1-{\Pr}\left\{\Gamma_{E}\leq\tau_e\right\}\notag\\
&=1-{\Pr}\left\{\max_{i\in\Phi} \left\{\max\left\{\gamma_{si},\gamma_{ri}\right\}\right\}\leq\tau_e\right\}\notag\\
&=1-\mathbb{E}_{\Phi}\left[\prod_{i\in\Phi}{\Pr}\left\{\max\left\{\gamma_{si},\gamma_{ri}\right\}\leq\tau_e\right\}\right]\notag\\
&=1-\mathbb{E}_{\Phi}\left[\prod_{i\in\Phi }\left(1-\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\right.\right.\notag\\
&~~\left.\left.\times\exp\left(-\frac{\tau_e}{\beta\overline{\gamma}_{si}}\right)\right)\left(1-\exp\left(-\frac{\tau_e}{\overline{\gamma}_{ri}}\right)\right)\right]\notag\\
&\overset{(a)}{=}1-\exp\left(-2\lambda\left(\mathcal{J}_1+\mathcal{J}_2-\mathcal{J}_3\right)\right).
%\overset{(a)}{=}&1-\exp\left[-2\lambda\int_0^{\infty}\int_{0}^{\pi}d_{si}\left(\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\right.\right.\notag\\
%&\left.\left.\times \exp\left(-\frac{\tau_e}{\beta P_s}d_{si}^{\eta}\right)+\exp\left(-\frac{\tau_e}{P_r}d_{ri}^{\eta}\right)\right.\right.\notag\\
%&\left.\left.-\exp\left(-\frac{\tau_e}{\beta P_s}d_{si}^{\eta}-\frac{\tau_e}{P_r}d_{ri}^{\eta}\right)\right.\right.\notag\\
%&\hspace{2cm}\left.\left.\times\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\right)dd_{si}d\theta\right]\notag\\
%=&1-\exp\left[-2\pi\lambda\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\right.\notag\\
%&\hspace{3cm}\left.\times\int_{0}^{\infty}d_{sj}\exp\left(-\frac{\tau_e}{\beta P_s}d_{sj}^{\eta}\right)dd_{sj}\right]\notag\\
%\overset{(b)}{=}&1-\exp\left[-\pi\lambda\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\right.\notag\\
%&\hspace{3cm}\left.\times\int_{0}^{\infty}\exp\left(-\frac{\tau_e}{\beta P_s}u^{\frac{\eta}{2}}\right)du\right]\notag\\
%\overset{(c)}{=}&1-\exp\left[-\frac{2\pi\lambda}{\eta}\left(\frac{\beta P_s}{\tau_e}\right)^{\frac{2}{\eta}}\right.\notag\\
%&\left.\times\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\int_0^{\infty}\exp\left(-t\right)t^{\frac{2}{\eta}-1}dt\right]\notag\\
%\overset{(d)}{=}&1-\exp\left[-\frac{2\pi\lambda}{\eta}\left(\frac{\beta P_s}{\tau_e}\right)^{\frac{2}{\eta}}\right.\notag\\
%&\hspace{1cm}\left.\times\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\Gamma\left(\frac{2}{\eta}\right)\right],
\end{align}
where
\begin{align}
\label{J_1} \mathcal{J}_1 = &\int_0^{\infty}\int_{0}^{\pi}d_{si}\left(\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\right.\notag\\
&\left.\hspace{2cm}\times
\exp\left(-\frac{\tau_e\sigma_{i1}^2}{\beta P_s}d_{si}^{\eta}\right)
\right)dd_{si}d\theta,
\end{align}
\begin{align}
\label{J_2} \mathcal{J}_2 =
\int_0^{\infty}\int_{0}^{\pi}d_{si}\exp\left(-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{ri}^{\eta}\right)dd_{si}d\theta,
\end{align}
and
\begin{align}
\label{J_3} \mathcal{J}_3 = & \int_0^{\infty}\int_{0}^{\pi}d_{si}\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\notag\\
&\hspace{1 cm}\times\exp\left(-\frac{\tau_e\sigma_{i1}^2}{\beta
P_s}d_{si}^{\eta}-\frac{\tau_e\sigma_{i2}^2}{P_r}d_{ri}^{\eta}\right)dd_{si}d\theta,
\end{align}
and $(a)$ follows by applying the probability generating functional
(PGFL) for the PPP $\Phi$, given by \cite{stoyan}
\begin{align}\label{pgfl}
\mathbb{E}_{\Phi}\left[\prod_{x\in\Phi}f\left(x\right)\right]=\exp\left\{-\int_{\mathbb{R}^2}\left[1-f\left(x\right)\right]\lambda
dx\right\},
\end{align}
and by changing to polar coordinates.

To proceed, we first derive $\mathcal{J}_1$ as
\begin{align}\label{J_1_2}
\mathcal{J}_1&=\pi\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\notag\\
&\hspace{2cm}\times\int_{0}^{\infty}d_{si}\exp\left(-\frac{\tau_e\sigma_{i1}^2}{\beta P_s}d_{si}^{\eta}\right)dd_{si}\notag\\
&\overset{(b)}{=}\frac{\pi}{2}\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\notag\\
&\hspace{2cm}\times\int_{0}^{\infty}\exp\left(-\frac{\tau_e\sigma_{i1}^2}{\beta P_s}u^{\frac{\eta}{2}}\right)du\notag\\
&\overset{(c)}{=}\frac{\pi}{\eta}\left(\frac{\beta P_s}{\tau_e\sigma_{i1}^2}\right)^{\frac{2}{\eta}}\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\notag\\
&\hspace{2cm}\times\int_0^{\infty}\exp\left(-t\right)t^{\frac{2}{\eta}-1}dt\notag\\
&\overset{(d)}{=}\frac{\pi}{\eta}\left(\frac{\beta
P_s}{\tau_e\sigma_{i1}^2}\right)^{\frac{2}{\eta}}
\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\Gamma\left(\frac{2}{\eta}\right),
\end{align}
where in $(b)$ we have used $u=d_{si}^2$, in $(c)$ we have used
$t=\frac{\tau_e\sigma_{i1}^2}{\beta P_s}u^{\frac{\eta}{2}}$, and
$(d)$ follows from the definition of the gamma function. We then
derive $\mathcal{J}_2$ as
\begin{align}\label{J_2_2}
&\mathcal{J}_2\notag\\
=&\int_{0}^{\infty}\!\!\!\!\int_{0}^{\pi}\!\!\!d_{si}\exp\!\left(\!\!-\frac{\tau_e\sigma_{i2}^2}{P_r}\!\left(d_{sr}^2+d_{si}^2\!-\!2d_{sr}d_{si}\cos\theta\right)^{\frac{\eta}{2}}\!\right)\!dd_{si}d\theta.
\end{align}
We further derive $\mathcal{J}_3$ as
\begin{align}\label{J_3_2}
&\mathcal{J}_3\notag\\
=&\left(1+\frac{\left(1-\beta\right)\tau_e}{\beta\left(N-1\right)}\right)^{-\left(N-1\right)}\int_{0}^{\infty}\int_{0}^{\pi}d_{si}\exp\left(-\frac{\tau_e\sigma_{i1}^2}{\beta P_s}d_{si}^{\eta}\right)\notag\\
&\times\exp\!\left(\!-\frac{\tau_e\sigma_{i2}^2}{P_r}\left(d_{sr}^2\!+\!d_{si}^2\!-\!2d_{sr}d_{si}\cos\theta\right)^{\frac{\eta}{2}}\!\right)\!dd_{si}d\theta.
\end{align}

Substituting \eqref{J_1_2}, \eqref{J_2_2}, and \eqref{J_3_2} into
\eqref{p_sec_2}, we obtain the desired result in \eqref{t1_result},
which completes the proof.
\end{appendices}

\begin{thebibliography}{10}

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\newblock {\em IEEE Signal Process. Mag.}, vol. 30, no. 5, pp. 29--40, Sep. 2013.

\bibitem{Yang_Mag}
N. Yang, L. Wang, G. Geraci, M. Elkashlan, J. Yuan, and M. Di
Renzo,``Safeguarding 5G wireless communication networks using
physical layer security," {\em IEEE Commun. Mag.}, accepted to
appear.

\bibitem{wyner}
A. Wyner,
\newblock ``The wire-tap channel,''
\newblock {\em Bell Syst. Tech. J.}, vol. 54, no. 8, pp. 1355--1387, Oct. 1975.

\bibitem{khisti}
A. Khisti and G. W. Wornell, ``Secure transmission with multiple
antennas \uppercase\expandafter{\romannumeral1}: The MISOME wiretap
channel,'' {\em IEEE Trans. Inf. Theory,} vol. 56, no. 6, pp.
3088--3104, Jul. 2010.

\bibitem{wornell}
A. Khisti and G. W. Wornell, ``Secure transmission with multiple
antennas \uppercase\expandafter{\romannumeral2}: The MIMOME wiretap
channel,'' {\em IEEE Trans. Inf. Theory,} vol. 56, no. 11, pp.
5515--5532, Nov. 2010.

\bibitem{chenxi}
C. Liu, G. Geraci, N. Yang, J. Yuan, and R. Malaney, ''Beamforming
for MIMO Gaussian channels with imperfect channel state
information,'' in {\em Proc. IEEE GlobeCOM 2013}, Atlanta, USA, Dec.
2013.

\bibitem{chenxi2}
C. Liu, N. Yang, G. Geraci, J. Yuan, and R. Malaney, ``Secrecy in
MIMOME wiretap channels: Beamforming with imperfect CSI,'' in {\em
Proc. IEEE ICC 2014}, Sydney, Australia, Jun. 2014.

\bibitem{chenxi3}
C. Liu, N. Yang, S. Yan, J. Yuan, and R. Malaney,``Secure adaptive
transmission in two-way relay wiretap channels,'' in {\em Proc.
IEEE/CIC ICCC 2014}, Shanghai, China, Oct. 2014.

\bibitem{Zhou10}
X. Zhou and M. R. McKay, B. Maham, and A. Hj{\o}rungnes,
\newblock ``Rethinking the secrecy outage formulation: A secure transmission design perspective,''
\newblock {\em IEEE Commun. Lett.}, vol. 15, no. 3, pp. 302--304, Mar. 2011.

\bibitem{Zhang13}
X. Zhang, X. Zhou, and M. R. McKay,
\newblock ``On the design of artificial-noise-aided secure multi-antenna transmission in slow fading channels,''
\newblock {\em IEEE Trans. Veh. Technol.}, vol. 62, no. 5, pp. 2170--2181, Jun. 2013.

\bibitem{nan}
N. Yang, P. L. Yeoh, M. Elkashlan, R. Schober, and I. B. Collings,
``Transmit antenna selection for security enhancement in MIMO
wiretap channels,'' {\em IEEE Trans. Commun.}, vol. 61, no. 1, pp.
144--154, Jan. 2013.

\bibitem{nan2}
N. Yang, H. A. Suraweera, I. B. Collings, and C. Yuen, ``Physical
layer security of TAS/MRC with antenna correlation,'' {\em IEEE
Trans. Inf. Forensic Security}, vol. 8, no. 1, pp. 254--259, Jan.
2013.

\bibitem{nan3}
N. Yang, P. L. Yeoh, M. Elkashlan, R. Schober, and J. Yuan, ``MIMO
wiretap channels: A secure transmission using transmit antenna
selection and receive generalized selection combining,'' {\em IEEE
Commun. Lett.}, vol. 17, no. 9, pp. 1754--1757, Sep. 2013.

\bibitem{nan4}
N. Yang, G. Geraci, J. Yuan, and R. Malaney, ``Confidential
broadcasting via linear precoding in non-homogeneous MIMO multiuser
networks,'' {\em IEEE Trans. Commun.}, vol. 62, no. 7, pp.
2515--2530, July 2014.

\bibitem{shihao}
S. Yan, N. Yang, R. Malaney, and J. Yuan, ``Transmit antenna
selection with Alamouti coding and power allocation in MIMO wiretap
channels,'' {\em IEEE Trans. Wireless Commun.}, vol. 13, no. 3, pp.
1656--1667, Mar. 2014.

\bibitem{xiangyun_2}
X. Zhou, R. K. Ganti, J. G. Andrews, and A. Hj{\o}rungnes, ``On the
throughput cost of physical layer security in decentralized wireless
networks,'' {\em IEEE Trans. Wireless Commun.}, vol. 10, no. 8, pp.
2764--2775, Aug. 2011.

\bibitem{wang_he}
H. Wang, X. Zhou, and M. C. Reed, ``Physical layer security in
cellular networks: A stochastic geometry approach,'' {\em IEEE
Trans. Wireless Commun.}, vol. 12, no. 6, pp. 2776--2787, Jun. 2013.

\bibitem{gio_1}
G. Geraci, S. Singh, J. G. Andrews, J. Yuan, and I. B. Collings,
``Secrecy rates in broadcast channels with confidential messages and
external eavesdroppers,''{\em IEEE Trans. Wireless Commun.}, vol.
13, no. 5, pp. 2931--2943, May 2014.

\bibitem{gio_2}
G. Geraci, H. S. Dhillon, J. G. Andrews, J. Yuan, and I. B.
Collings, ``Physical layer security in downlink multi-antenna
cellular networks,'' {\em IEEE Trans. Commun.}, vol. 62, no. 6, pp.
2006--2021, Jun. 2014.

%\bibitem{huiming} T.-X. Zheng, H.-M. Wang, J. Yuan, D. Towsley, and M. H. Lee, ``Multi-antenna transmission with artificial noise against randomly distributed eavesdroppers,'' {\em IEEE Trans. Commun.}, accepted to appear.


\bibitem{Laneman}
J. N. Laneman, D. N. C. Tse, and G. W. Wornell, ``Cooperative
diversity in wireless networks: Efficient protocols and outage
behavior,'' {\em IEEE Trans. Inf. Theory}, vol. 50, pp. 3062--3080,
Dec. 2004.

\bibitem{Bassily}
R. Bassily, E. Ekrem, X. He, E. Tekin, J. Xie, M. R. Bloch, S.
Ulukus, and A. Yener,
\newblock ``Cooperative security at the physical layer: A summary of recent advances,''
\newblock {\em IEEE Signal Process. Mag.}, vol. 30, no. 5, pp. 16--28, Sep. 2013.

\bibitem{dong}
L. Dong, Z. Han, A. Petropulu, and H. V. Poor, ``Improving wireless
physical layer security via cooperating relays,'' {\em IEEE Trans.
Signal Process.}, vol. 58, no. 3, pp. 1875--1888, Mar. 2010.

\bibitem{xiaoming_1} X. Chen, C. Zhong, C. Yuen, and H.-H. Chen, ``Multi-antenna relay aided wireless physical layer security,'' {\em IEEE Commun. Mag.}, accepted.

\bibitem{xiaoming_2} X. Chen, L. Lei, H. Zhang, and C. Yuen, ``Large-scale MIMO relaying techniques for physical layer security: AF or DF?'' {\em IEEE Trans. Wireless Commun.}, accepted.

\bibitem{yulong}
Y. Zou, X. Wang, and W. Shen, ``Optimal relay selection for
physical-layer security in cooperative wireless networks,'' {\em
IEEE J. Sel. Areas Commun.}, vol. 31, no. 10, pp. 2099--2111, Oct.
2013.

\bibitem{hanzhu}
J. Chen, L. Song, Z. Han, and B. Jiao, ``Joint relay and jammer
selection for secure two-way relay networks,'' {\em IEEE Trans. Inf.
Foren. Sec.}, vol. 7, no. 1, pp. 310--320, Feb. 2012.

\bibitem{chenxi4}
C. Liu, N. Yang, J. Yuan, and R. Malaney, ``Location-based secure
transmission for wiretap channels,'' {\em IEEE J. Sel. Areas
Commun.}, vol. 33, no. 7, pp. 1458--1470, Jul. 2015.

\bibitem{weber} S. Weber, J. G. Andrews, and N. Jindal, ``An overview of the transmission capacity of wireless networks,'' {\em IEEE Trans. Commun.}, vol. 58, no. 12, pp. 3593--3604, Dec. 2010.
    
\bibitem{chenxi5} C. Liu, N. Yang, R. Malaney, and J. Yuan, ``Artificial-noise-aided transmission in multi-antenna relay wiretap channels with spatially random eavesdroppers,'' arXiv:1509.05486.

\bibitem{stoyan}
D. Stoyan, W. Kendall, and J. Mecke, {\em Stochastic Geometry and
its Applications}, 2nd ed. John Wiley $\&$ Sons Ltd., 1996.

%\bibitem{table}
%I. S. Gradshteyn and I. M. Ryzhik, {\em Tabel of Integrals, Series,
%and Products,} 7th edition. Academic Press, 2007.

\end{thebibliography}

\end{document}

1509.07223.extracted_bib
{Hong Y.-W. P. Hong, P.-C. Lan, and C.-C. J. Kuo, \newblock ``Enhancing physical-layer secrecy in multiantenna wireless systems: An overview of signal processing approaches,'' \newblock {\em IEEE Signal Process. Mag., vol. 30, no. 5, pp. 29--40, Sep. 2013.
1509.07223.extracted_bib
{Yang_Mag N. Yang, L. Wang, G. Geraci, M. Elkashlan, J. Yuan, and M. Di Renzo,``Safeguarding 5G wireless communication networks using physical layer security," {\em IEEE Commun. Mag., accepted to appear.
1509.07223.extracted_bib
{wyner A. Wyner, \newblock ``The wire-tap channel,'' \newblock {\em Bell Syst. Tech. J., vol. 54, no. 8, pp. 1355--1387, Oct. 1975.
1509.07223.extracted_bib
{khisti A. Khisti and G. W. Wornell, ``Secure transmission with multiple antennas \uppercase\expandafter{\romannumeral1: The MISOME wiretap channel,'' {\em IEEE Trans. Inf. Theory, vol. 56, no. 6, pp. 3088--3104, Jul. 2010.
1509.07223.extracted_bib
{wornell A. Khisti and G. W. Wornell, ``Secure transmission with multiple antennas \uppercase\expandafter{\romannumeral2: The MIMOME wiretap channel,'' {\em IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5515--5532, Nov. 2010.
1509.07223.extracted_bib
{chenxi C. Liu, G. Geraci, N. Yang, J. Yuan, and R. Malaney, ''Beamforming for MIMO Gaussian channels with imperfect channel state information,'' in {\em Proc. IEEE GlobeCOM 2013, Atlanta, USA, Dec. 2013.
1509.07223.extracted_bib
{chenxi2 C. Liu, N. Yang, G. Geraci, J. Yuan, and R. Malaney, ``Secrecy in MIMOME wiretap channels: Beamforming with imperfect CSI,'' in {\em Proc. IEEE ICC 2014, Sydney, Australia, Jun. 2014.
1509.07223.extracted_bib
{chenxi3 C. Liu, N. Yang, S. Yan, J. Yuan, and R. Malaney,``Secure adaptive transmission in two-way relay wiretap channels,'' in {\em Proc. IEEE/CIC ICCC 2014, Shanghai, China, Oct. 2014.
1509.07223.extracted_bib
{Zhou10 X. Zhou and M. R. McKay, B. Maham, and A. Hj{\orungnes, \newblock ``Rethinking the secrecy outage formulation: A secure transmission design perspective,'' \newblock {\em IEEE Commun. Lett., vol. 15, no. 3, pp. 302--304, Mar. 2011.
1509.07223.extracted_bib
{Zhang13 X. Zhang, X. Zhou, and M. R. McKay, \newblock ``On the design of artificial-noise-aided secure multi-antenna transmission in slow fading channels,'' \newblock {\em IEEE Trans. Veh. Technol., vol. 62, no. 5, pp. 2170--2181, Jun. 2013.
1509.07223.extracted_bib
{nan N. Yang, P. L. Yeoh, M. Elkashlan, R. Schober, and I. B. Collings, ``Transmit antenna selection for security enhancement in MIMO wiretap channels,'' {\em IEEE Trans. Commun., vol. 61, no. 1, pp. 144--154, Jan. 2013.
1509.07223.extracted_bib
{nan2 N. Yang, H. A. Suraweera, I. B. Collings, and C. Yuen, ``Physical layer security of TAS/MRC with antenna correlation,'' {\em IEEE Trans. Inf. Forensic Security, vol. 8, no. 1, pp. 254--259, Jan. 2013.
1509.07223.extracted_bib
{nan3 N. Yang, P. L. Yeoh, M. Elkashlan, R. Schober, and J. Yuan, ``MIMO wiretap channels: A secure transmission using transmit antenna selection and receive generalized selection combining,'' {\em IEEE Commun. Lett., vol. 17, no. 9, pp. 1754--1757, Sep. 2013.
1509.07223.extracted_bib
{nan4 N. Yang, G. Geraci, J. Yuan, and R. Malaney, ``Confidential broadcasting via linear precoding in non-homogeneous MIMO multiuser networks,'' {\em IEEE Trans. Commun., vol. 62, no. 7, pp. 2515--2530, July 2014.
1509.07223.extracted_bib
{shihao S. Yan, N. Yang, R. Malaney, and J. Yuan, ``Transmit antenna selection with Alamouti coding and power allocation in MIMO wiretap channels,'' {\em IEEE Trans. Wireless Commun., vol. 13, no. 3, pp. 1656--1667, Mar. 2014.
1509.07223.extracted_bib
{xiangyun_2 X. Zhou, R. K. Ganti, J. G. Andrews, and A. Hj{\orungnes, ``On the throughput cost of physical layer security in decentralized wireless networks,'' {\em IEEE Trans. Wireless Commun., vol. 10, no. 8, pp. 2764--2775, Aug. 2011.
1509.07223.extracted_bib
{wang_he H. Wang, X. Zhou, and M. C. Reed, ``Physical layer security in cellular networks: A stochastic geometry approach,'' {\em IEEE Trans. Wireless Commun., vol. 12, no. 6, pp. 2776--2787, Jun. 2013.
1509.07223.extracted_bib
{gio_1 G. Geraci, S. Singh, J. G. Andrews, J. Yuan, and I. B. Collings, ``Secrecy rates in broadcast channels with confidential messages and external eavesdroppers,''{\em IEEE Trans. Wireless Commun., vol. 13, no. 5, pp. 2931--2943, May 2014.
1509.07223.extracted_bib
{gio_2 G. Geraci, H. S. Dhillon, J. G. Andrews, J. Yuan, and I. B. Collings, ``Physical layer security in downlink multi-antenna cellular networks,'' {\em IEEE Trans. Commun., vol. 62, no. 6, pp. 2006--2021, Jun. 2014. %
1509.07223.extracted_bib
{huiming T.-X. Zheng, H.-M. Wang, J. Yuan, D. Towsley, and M. H. Lee, ``Multi-antenna transmission with artificial noise against randomly distributed eavesdroppers,'' {\em IEEE Trans. Commun., accepted to appear.
1509.07223.extracted_bib
{Laneman J. N. Laneman, D. N. C. Tse, and G. W. Wornell, ``Cooperative diversity in wireless networks: Efficient protocols and outage behavior,'' {\em IEEE Trans. Inf. Theory, vol. 50, pp. 3062--3080, Dec. 2004.
1509.07223.extracted_bib
{Bassily R. Bassily, E. Ekrem, X. He, E. Tekin, J. Xie, M. R. Bloch, S. Ulukus, and A. Yener, \newblock ``Cooperative security at the physical layer: A summary of recent advances,'' \newblock {\em IEEE Signal Process. Mag., vol. 30, no. 5, pp. 16--28, Sep. 2013.
1509.07223.extracted_bib
{dong L. Dong, Z. Han, A. Petropulu, and H. V. Poor, ``Improving wireless physical layer security via cooperating relays,'' {\em IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1875--1888, Mar. 2010.
1509.07223.extracted_bib
{xiaoming_1 X. Chen, C. Zhong, C. Yuen, and H.-H. Chen, ``Multi-antenna relay aided wireless physical layer security,'' {\em IEEE Commun. Mag., accepted.
1509.07223.extracted_bib
{xiaoming_2 X. Chen, L. Lei, H. Zhang, and C. Yuen, ``Large-scale MIMO relaying techniques for physical layer security: AF or DF?'' {\em IEEE Trans. Wireless Commun., accepted.
1509.07223.extracted_bib
{yulong Y. Zou, X. Wang, and W. Shen, ``Optimal relay selection for physical-layer security in cooperative wireless networks,'' {\em IEEE J. Sel. Areas Commun., vol. 31, no. 10, pp. 2099--2111, Oct. 2013.
1509.07223.extracted_bib
{hanzhu J. Chen, L. Song, Z. Han, and B. Jiao, ``Joint relay and jammer selection for secure two-way relay networks,'' {\em IEEE Trans. Inf. Foren. Sec., vol. 7, no. 1, pp. 310--320, Feb. 2012.
1509.07223.extracted_bib
{chenxi4 C. Liu, N. Yang, J. Yuan, and R. Malaney, ``Location-based secure transmission for wiretap channels,'' {\em IEEE J. Sel. Areas Commun., vol. 33, no. 7, pp. 1458--1470, Jul. 2015.
1509.07223.extracted_bib
{weber S. Weber, J. G. Andrews, and N. Jindal, ``An overview of the transmission capacity of wireless networks,'' {\em IEEE Trans. Commun., vol. 58, no. 12, pp. 3593--3604, Dec. 2010.
1509.07223.extracted_bib
{chenxi5 C. Liu, N. Yang, R. Malaney, and J. Yuan, ``Artificial-noise-aided transmission in multi-antenna relay wiretap channels with spatially random eavesdroppers,'' arXiv:1509.05486.
1509.07223.extracted_bib
{stoyan D. Stoyan, W. Kendall, and J. Mecke, {\em Stochastic Geometry and its Applications, 2nd ed. John Wiley $\&$ Sons Ltd., 1996. %
1509.07223.extracted_bib
{table %I. S. Gradshteyn and I. M. Ryzhik, {\em Tabel of Integrals, Series, %and Products, 7th edition. Academic Press, 2007.