text:
\section*{Appendix A
\renewcommand\theequation{A.\arabic{equation}}
\setcounter{equation}{0}
\begin{center}
Proof of Lemma \ref{thm:ancoutage}
\end{center}

Let $X=|h|^2$ and $Y=|f|^2$. Thus, $X$ and $Y$ are exponential
random variables with means $\Omega_h$ and $\Omega_f$,
respectively. By using the inequality (\ref{eqn:hormonic}), the
outage probability can be lower-bounded as follows:
\begin{eqnarray}
P_{\ANC}^{\outage}(R) &=& \Pr\left(I_1^\ANC<\frac{R}{2} \ \ {\mathrm{or}} \ \ I_2^\ANC<\frac{R}{2}\right)\\
&=&1-\Pr\left(I_1^\ANC>\frac{R}{2}, I_2^\ANC>\frac{R}{2}\right)\\
&=&1-\Pr\left(\gamma_1^\ANC>2^{R}-1, \gamma_1^\ANC<\gamma_2^\ANC
\right)\nonumber\\
&&-\Pr\left(\gamma_2^\ANC>2^{R}-1,
\gamma_2^\ANC<\gamma_1^\ANC \right)\\
&>&1-\Pr\left(\frac{E}{2}\min(2X,Y)>2^{R}-1,\min(2X,Y)<\min(2Y,X)\right)\nonumber\\
&&-\Pr\left(\frac{E}{2}\min(2Y,X)>2^{R}-1,\min(2Y,X)<\min(2X,Y)\right).\label{eqn:outANC1}
\end{eqnarray}
The first probability in (\ref{eqn:outANC1}) can be evaluated in
the following way
\begin{eqnarray}
&&\Pr\left(\frac{E}{2}\min(2X,Y)>2^{R}-1,\min(2X,Y)<\min(2Y,X)\right)\nonumber\\
&=&\Pr\left(Y>\frac{2\left(2^{R}-1\right)}{E},\min(2Y,X)>Y,2X>Y\right)\\
&=&\Pr\left(Y>\frac{2\left(2^{R}-1\right)}{E},X>2Y\right
+\Pr\left(Y>\frac{2\left(2^{R}-1\right)}{E},2Y>X>Y\right)\\
&=&\Pr\left(Y>\frac{2\left(2^{R}-1\right)}{E},X>Y\right)\\
&=&\frac{\Omega_h}{\Omega_h+\Omega_f}\exp\left(
-\frac{2\left(\Omega_h+\Omega_f\right)\left(2^{R}-1\right)}{E\Omega_h\Omega_f}\right),\label{eqn:outANCp1}
\end{eqnarray}
where the integrations involved in the last step can be solved by
\cite{int}. Similarly, the second probability in
(\ref{eqn:outANC1}) can be solved as follows:
\begin{eqnarray}\label{eqn:outANCp2}
&&\Pr\left(\frac{E}{2}\min(2Y,X)>2^{R}-1,\min(2Y,X)<\min(2X,Y)\right)\nonumber\\
&=&\frac{\Omega_f}{\Omega_h+\Omega_f}\exp\left(
-\frac{2\left(\Omega_h+\Omega_f\right)\left(2^{R}-1\right)}{E\Omega_h\Omega_f}\right).
\end{eqnarray}
By substituting (\ref{eqn:outANCp1}) and (\ref{eqn:outANCp2}) into
(\ref{eqn:outANC1}), we obtain the lower bound
$\tilde{P}_\ANC^\outage(R)$ of the outage probability
$P_\ANC^\outage(R)$.

\section*{Appendix B
\renewcommand\theequation{B.\arabic{equation}}
\setcounter{equation}{0}
\begin{center}
Proof of Lemma \ref{thm:outTDBC}
\end{center}

Let $X=|h|^2$, $Y=|f|^2$, and $Z=|g|^2$. Thus, $X$, $Y$, and $Z$
are exponential random variables with means $\Omega_h$,
$\Omega_f$, and $\Omega_g$, respectively. By using the inequality
in (\ref{eqn:hormonic}), the outage probability is lower-bounded
by
\begin{eqnarray}
P_\TDBC^\outage(R) &=& \Pr\left(I_1^\TDBC<\frac{R}{2} \ \
\mathrm{or} \ \
I_2^\TDBC<\frac{R}{2}\right)\\
&=&1-\Pr\left(I_1^\TDBC>\frac{R}{2}, I_2^\TDBC>\frac{R}{2}\right)\\
&=&1-\Pr\left(\gamma_1^\TDBC>2^{1.5R}-1,
\gamma_1^\TDBC<\gamma_2^\TDBC
\right)\nonumber\\
&&-\Pr\left(\gamma_2^\TDBC>2^{1.5R}-1,
\gamma_2^\TDBC<\gamma_1^\TDBC \right)\\
&>&1-\Pr\left(\frac{E}{3}\min(3Y,X)+EZ>2^{1.5R}-1,\min(3Y,X)<\min(3X,Y)\right)\nonumber\\
&&-\Pr\left(\frac{E}{3}\min(3X,Y)+EZ>2^{1.5R}-1,\min(3X,Y)<\min(3Y,X)\right).\label{eqn:outTDBC1}
\end{eqnarray}

The first probability in (\ref{eqn:outTDBC1}) can be solved as
follows:
\begin{eqnarray}
&&\Pr\left(\frac{E}{3}\min(3Y,X)+EZ>2^{1.5R}-1,\min(3Y,X)<\min(3X,Y)\right)\nonumber\\
&=&\Pr\left(\frac{1}{3}X+Z>\frac{2^{1.5R}-1}{E},X<\min(3X,Y),X<3Y\right)\nonumber\\
&&+\Pr\left(aY+Z>\frac{2^{1.5R}-1}{E},3Y<\min(3X,Y),X>3Y\right)\\
&=&\Pr\left(\frac{1}{3}X+Z>\frac{2^{1.5R}-1}{E},X<3Y,3X<Y\right)\nonumber\\
&&+\Pr\left(\frac{1}{3}X+Z>\frac{2^{1.5R}-1}{E},X<Y,X<3Y,3X>Y\right)\\
&=&\Pr\left(\frac{1}{3}X+Z>\frac{2^{1.5R}-1}{E},X<\frac{Y}{3}\right)\nonumber\\
&&+\Pr\left(\frac{1}{3}X+Z>\frac{2^{1.5R}-1}{E},\frac{Y}{3}<X<Y\right)\\
&=&\Pr\left(\frac{1}{3}X+Z>\frac{2^{1.5R}-1}{E},X<Y\right)\\
&=&\frac{\Omega_f}{(3\Omega_f\Omega_g+3\Omega_h\Omega_g-\Omega_h\Omega_f)(\Omega_h+\Omega_f)}\left(
3\Omega_g\left(\Omega_f+\Omega_h\right)\exp\left(-\frac{2^{1.5R}-1}{E\Omega_g}\right)\right.\nonumber\\
&&\left.-\Omega_h\Omega_f\exp\left(-\frac{3(2^{1.5R}-1)(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f}\right)
\right).\label{eqn:tdbcprob1}
\end{eqnarray}
where the integrations involved in the last step can be solved by
\cite{int}. Similarly, the second integration can be solved in the
following way
\begin{eqnarray}
&&\Pr\left(\frac{E}{3}\min(3X,Y)+EZ>2^{1.5R}-1,\min(3X,Y)<\min(3Y,X)\right)\nonumber\\
&=&\frac{\Omega_h}{(3\Omega_f\Omega_g+3\Omega_h\Omega_g-\Omega_h\Omega_f)(\Omega_h+\Omega_f)}\left(
3\Omega_g\left(\Omega_f+\Omega_h\right)\exp\left(-\frac{2^{1.5R}-1}{E\Omega_g}\right)\right.\nonumber\\
&&\left.-\Omega_h\Omega_f\exp\left(-\frac{3(2^{1.5R}-1)(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f}\right)
\right).\label{eqn:tdbcprob2}
\end{eqnarray}
Based on (\ref{eqn:outTDBC1}), (\ref{eqn:tdbcprob1}), and
(\ref{eqn:tdbcprob2}), the outage probability of the TDBC protocol
is lower-bounded by $\tilde{P}_\TDBC^\outage(R)$.

\section*{Appendix C
\renewcommand\theequation{C.\arabic{equation}}
\setcounter{equation}{0}
\begin{center}
Proof of Theorem \ref{thm:ancpower1}
\end{center}

We let $E_1=3\alpha E$, $E_2=3\beta E$, and $E_r =
3(1-\alpha-\beta)E$, where $0\leq \alpha\leq 1$, $0\leq \beta\leq
1$, and $\alpha+\beta\leq 1$. Consequently, $\gamma_1^\ANC$ in and
$\gamma_2^\ANC$ in (\ref{eqn:ancsnr12gen}) are approximated by
\begin{eqnarray}
\gamma_1^\ANC \approx
3|hf|^2E\frac{\beta(1-\alpha-\beta)}{|h|^2(1-\beta)+|f|^2\beta},&&
\gamma_2^\ANC \approx
3|hf|^2E\frac{\alpha(1-\alpha-\beta)}{|f|^2(1-\alpha)+|h|^2\alpha}.
\end{eqnarray}
We follow the method given in Section II.C of \cite{poor} to solve
the minimax problem in (\ref{eqn:ancminmax}). Thus, a new function
$M(\pi_0,\alpha,\beta)$ is defined as
\begin{equation}
M(\pi_0,\alpha,\beta) =
\pi_0\gamma_1^\ANC+(1-\pi_0)\gamma_2^\ANC,\ \ \ 0\leq \pi_0\leq 1.
\end{equation}
Let $\alpha(\pi_0)$ and $\beta(\pi_0)$ denote the values of
$\alpha$ and $\beta$, respectively, which maximize
$M(\pi_0,\alpha,\beta)$ for a fixed $\pi_0$. According to
\cite{poor}, the solution to (\ref{eqn:ancminmax}) must belong to
the set formed by $\alpha(\pi_0)$ and $\beta(\pi_0)$, i.e. it must
maximize $M(\pi_0,\alpha,\beta)$.

We try to find $\alpha(\pi_0)$ and $\beta(\pi_0)$ in the
following. When $\pi_0\neq 1/2$, it can be shown that the values
of $\alpha(\pi_0)$ and $\beta(\pi_0)$ are given by
\begin{eqnarray}\label{eqn:nothalf}
\left\{%
\begin{array}{ccl}
  \alpha(\pi_0) = 0, &
  \beta(\pi_0)=\frac{|h|}{|h|+|f|}, & {\mathrm{when}}\ 0\leq\pi_0<1/2\\
  \beta(\pi_0) = 0, &
  \alpha(\pi_0)=\frac{|f|}{|h|+|f|}, & {\mathrm{when}}\ 1/2<\pi_0\leq 1
\end{array}%
\right..
\end{eqnarray}
When either $\alpha=0$ or $\beta=0$, however, it follows from
(\ref{eqn:ancr1r2}) and (\ref{eqn:ancout}) that the outage
probability is one. This implies that the solution in
(\ref{eqn:nothalf}) can not be the solution to
(\ref{eqn:ancminmax}).

Consequently, the solution to (\ref{eqn:ancminmax}) can only be
found at $\pi_0=1/2$. For this case, we first find that
$\alpha(1/2)$ and $\beta(1/2)$ are not unique. Specifically,
$\beta(1/2)$ can be any real number between zero and one, and the
value of $\alpha(1/2)$ depends on $\beta(1/2)$ in the following
way
\begin{eqnarray}\label{eqn:alpha1}
\alpha(1/2) =
\frac{|f|\left(|hf|-(|f|^2\beta+|h|^2(1-\beta(1/2))\right)}{|h|(|f|^2-|h|^2)}.
\end{eqnarray}
On the other hand, it has been shown in \cite{poor} that the
solution to (\ref{eqn:ancminmax})
should make $\gamma_1^\ANC=\gamma_2^\ANC$.
Based on this condition and (\ref{eqn:alpha1}), we should choose
$\alpha(1/2)$ and $\beta(1/2)$ as follows:
\begin{eqnarray}
\alpha(1/2) = \frac{|f|}{2(|h|+|f|)},&& \beta(1/2) =
\frac{|h|}{2(|h|+|f|)}.
\end{eqnarray}
Therefore, the optimum power allocation scheme that minimizes the
outage probability of the ANC protocol is given by
(\ref{eqn:ancer})--(\ref{eqn:ance2}).

\section*{Appendix D
\renewcommand\theequation{D.\arabic{equation}}
\setcounter{equation}{0}
\begin{center}
Proof of Theorem \ref{thm:tdbcout}
\end{center}

In this proof, we rewrite $\gamma_1^\TDBC$ and $\gamma_2^\TDBC$ as
$\gamma_1^\TDBC(\xi)$ and $\gamma_2^\TDBC(\xi)$ in order to
emphasize their dependence on $\xi$. Furthermore, we construct a
new function $W(\pi_0,\xi)$ as follows:
\begin{equation}
W(\pi_0,\xi) =
\pi_0\gamma_1^\TDBC(\xi)+(1-\pi_0)\gamma_2^\TDBC(\xi),\ \ \ 0\leq
\pi_0\leq 1.
\end{equation}
Let $\xi(\pi_0)$ denote the value of $\xi$ that maximizes
$W(\pi_0,\xi)$ for a fixed $\pi_0$. Let
$V(\pi_0)=W(\pi_0,\xi(\pi_0))$ and let $\pi_L$ denote the solution
to $\arg\min_{0\leq \pi_0\leq 1} V(\pi_0)$. It has been shown that
the solution to (\ref{eqn:tdbcminmax2}) must be from the set
formed by $\xi(\pi_0)$ \cite{poor}. Furthermore, it follows from
Proposition II.C.1 in \cite{poor} that $\xi(\pi_L)$ is the
solution to (\ref{eqn:tdbcminmax2}) either
$\gamma_1^\TDBC(\xi(\pi_L))=\gamma_2^\TDBC(\xi(\pi_L))$,
$\pi_L=0$, or $\pi_L=1$.

As a result, in order to find the optimum value of $\xi$, we can
first investigate the equation
$\gamma_1^\TDBC(\xi)=\gamma_2^\TDBC(\xi)$. If a solution exits for
this equation, such solution must be the solution to
(\ref{eqn:tdbcminmax2}). Fortunately, the equation
$\gamma_1^\TDBC(\xi)=\gamma_2^\TDBC(\xi)$ always have a solution
under the assumption that every terminal has the same transmission
power $E$. Specifically, it can be easily shown that
$\gamma_1^\TDBC(\xi)$ is a strictly decreasing function of $\xi$;
while, $\gamma_2^\TDBC(\xi)$ is a strictly increasing function of
$\xi$. Furthermore, $\gamma_1^\TDBC(0)=\gamma_2^\TDBC(1)$ and
$\gamma_1^\TDBC(1)=\gamma_2^\TDBC(0)$. As result, the two
functions $\gamma_1^\TDBC(\xi)$ and $\gamma_2^\TDBC(\xi)$ always
have one and only one crossing point in the range $0\leq \xi \leq
1$, i.e. the equation $\gamma_1^\TDBC(\xi)=\gamma_2^\TDBC(\xi)$
always has a solution. Such solution is given by $|h|/(|h|+ |f|)$,
and hence, the solution to (\ref{eqn:tdbcminmax2}) is given by
(\ref{eqn:xitdbc1}).

\section{Introduction}\label{sec:intro}

In traditional unidirectional cooperative networks, several relays
assist in the communication between one source and one destination
in order to achieve spatial diversity \cite{laneman1}. The {\it
diversity-multiplexing tradeoff}, which is one of the most
fundamental properties of any communication systems, of such
networks has been extensively studied
\cite{azarian}--\cite{yuksel}. It has been shown that, in the
unidirectional cooperative networks, the half-duplex constraint of
every terminal induces a severe loss of bandwidth efficiency as
demonstrated by a pre-log factor $1/2$ in the mutual information
expression. In order to overcome this difficulty, bidirectional
cooperative networks were studied in \cite{rankov}, where two
sources exchanged information with the help of several relays. As
a result, there were two traffic flows in a bidirectional
cooperative network and they were supported by the same physical
channels concurrently. Although each traffic flow still had the
pre-log factor $1/2$ in its mutual information expression, the
total mutual information of the network, which was the summation
of the mutual information of both traffic flows, no longer
suffered from the pre-log factor $1/2$. Therefore, bidirectional
cooperative networks had much higher bandwidth efficiency than
unidirectional cooperative networks.

Recently, many novel protocols were studied in the context of
bidirectional cooperative networks, such as {\it physical layer
network coding} (PNC) \cite{shengli1}--\cite{rankov2}, {\it analog
network coding} (ANC) \cite{katti2}, \cite{shengli2}, and {\it
time division broadcast} (TDBC) \cite{shengli1} and
\cite{kim1}--\cite{liu}\footnote{Note that the TDBC protocol was
called the {\it straightforward network coding} scheme in
\cite{shengli1}.}. Those protocols not only achieved high
bandwidth efficiency but also successfully controlled the
interferences between the two traffic flows in a bidirectional
cooperative network. Many previous works assumed that there was no
direct channel between the two sources \cite{shengli1},
\cite{katti2}, \cite{shengli2}, and \cite{liu}--\cite{haoy}. Under
this assumption, the authors showed that the PNC and ANC protocols
had larger capacity regions and higher sum-rates than the TDBC
protocol \cite{shengli1}, \cite{katti2}, and \cite{haoy}. On the
other hand, since there is no direct channel, the maximum
diversity gains of the PNC, ANC, and TDBC protocols are the same,
just one. Therefore, one may conclude that the PNC and ANC
protocols always outperform the TDBC protocol in terms of
diversity-multiplexing tradeoff \cite{zheng}.

In fact, numerous previous publications on cooperative networks
have also considered the case that there is a direct channel
between the two sources \cite{laneman1}--\cite{laneman2},
\cite{yang}, \cite{yuksel}, \cite{kim1}, \cite{kim2}, and
\cite{anghel1}--\cite{katti}. For this case, the comparison of
diversity-multiplexing tradeoff might have different result. It is
not hard to see that the TDBC protocol can utilize this direct
channel \cite{kim1}, \cite{kim2}; but the PNC and ANC protocols
can not do so due to the half-duplex constraint \cite{katti}.
Thus, the TDBC protocol might have a higher diversity gain than
the PNC and ANC protocols. This is fundamentally different from
the case where the direct channel does not exist. Therefore, it is
necessary and interesting to compare those protocols in terms of
diversity-multiplexing tradeoff under the assumption that the
direct channel exists. Furthermore, it is more desirable to
compare the tradeoff at finite signal-to-noise ratio (SNR) range
as in \cite{nara} instead of only at infinite SNR in \cite{zheng},
because practical communication systems, such as wireless local
area networks, usually work in the SNR range $3$--$20$ dB.
However, such comparison has not been investigated in previous
publications. This has motivated our work.

In this paper, we assume that the relays work in the
amplify-and-forward mode, and hence, we do not consider the PNC
protocol. In fact, it has been shown that the PNC protocol may
suffer considerable performance loss in fading channels
\cite{haoy}, \cite{dingz}. Thus, we focus on the ANC and TDBC
protocols in this paper. The contributions of this paper are
summarized as follows:

\begin{itemize}
    \item We derive lower bounds of the
    outage probabilities of the ANC and TDBC protocols. Those bounds are extremely tight in the whole SNR range,
    irrespective of the values of channel variances.

    \item We obtain the finite-SNR diversity-multiplexing
    tradeoffs of the ANC and TDBC protocols. Those tradeoffs
    establish a framework which enables us to make a comprehensive comparison
    between those two protocols. For example, the maximum
    diversity gain of the TDBC protocol is twice larger than that of
    the ANC protocol. However, the ANC protocol achieves a higher diversity
    gain when the multiplexing gain is larger than $1/2$, irrespective
    of the value of SNR.

    \item For the ANC protocol, we propose an optimum power allocation scheme which
    simultaneously minimizes the outage probability and maximizes
    the total mutual information of this protocol.

    \item For the TDBC protocol, we develop an optimum method to combine the received signals at the relay when
    equal power allocation is applied. This method simultaneously minimizes the
    outage probability and maximizes the total mutual information as well.
\end{itemize}

The rest of this paper is organized as follows. Section
\ref{sec:sys} describes the system models of the ANC and TDBC
protocols. Section \ref{sec:prob} focuses on the outage
probabilities and finite-SNR diversity-multiplexing tradeoffs of
those two protocols. In Section \ref{sec:power}, we first develop
an optimum power allocation scheme for the ANC protocol. Then we
propose an optimum method for the TDBC protocol to combine the
received signals at the relay. Section \ref{sec:numeric} presents
some numerical results and Section \ref{sec:conl} concludes this
paper.

\section{System Model}\label{sec:sys}

We consider a bidirectional cooperative network with two sources
and one relay, where the sources intend to exchange information
with the help of the relay. Every terminal has only one antenna
and is half-duplex. We use $\rmS_1$, $\rmS_2$, and $\rmR$ to
denote the first source, the second source, and the relay,
respectively. Let $g$ represent the fading coefficient of the
channel between $\rmS_1$ and $\rmS_2$, $h$ the channel between
$\mathrm{S}_1$ and $\mathrm{R}$, and $f$ the channel between
$\rmR$ and $\mathrm{S}_2$. Furthermore, we assume that $g$, $h$,
and $f$ are complex Gaussian random variables with zero mean and
variances $\Omega_g$, $\Omega_h$, and $\Omega_f$, respectively.
The additive noise associated with every channel is assumed to be
a complex Gaussian random variable with zero mean and unit
variance. Let $x_1$ and $x_2$ denote the information-bearing
symbol transmitted from $\rmS_1$ and $\rmS_2$, respectively. Both
$x_1$ and $x_2$ have unit power. The total transmission power of
the bidirectional cooperative network is constrained to be $3E$.

Since the two sources intend to exchange information, there are
two traffic flows in this bidirectional cooperative network. One
is from $\rmS_1$ via $\rmR$ to $\rmS_2$ and the other is from
$\rmS_2$ via $\rmR$ to $\rmS_1$. Each traffic flow can be seen as
a traditional unidirectional cooperative network. For example, in
the first traffic flow, $\rmS_1$ is the transmitter, and $\rmR$
and $\rmS_2$ are the receivers. Hence, it is reasonable to assume
that $\rmR$ knows $h$ and $\rmS_2$ knows $h$, $f$, and $g$ as in
the conventional unidirectional cooperative networks
\cite{laneman1}, \cite{anghel1}. Similarly, due to the existence
of the second traffic flow, it is reasonable to assume that $\rmR$
knows $f$ and $\rmS_1$ knows $h$, $f$, and $g$. In all, we assume
that the two sources know $h$, $f$, and $g$, and the relay knows
$h$ and $f$ as in many previous publications \cite{rankov},
\cite{lee1}, \cite{lee2}, and \cite{kuhn}.

\subsection{Analog Network Coding (ANC)}

The ANC protocol has received lots of attention recently
\cite{katti2}, \cite{shengli2}, and it is illustrated in Fig.\
\ref{fig:model}(a). In this protocol, $\rmS_1$ and $\rmS_2$
simultaneously transmit to $\rmR$ at the first time slot with
power $E$. Thus, the received signal $y_R(1)$ of $\rmR$ at the
first time slot is given by
\begin{equation}
y_R(1) = \sqrt{E}hx_1+\sqrt{E}fx_2+n_R(1),
\end{equation}
where $n_R(1)$ is the additive Gaussian noise. At the second time
slot, $\rmR$ amplifies $y_R(1)$ with an amplifying coefficient
$\rho$ and then transmits it to $\rmS_1$ and $\rmS_2$ also with
power $E$.\footnote{The setting that every terminal has the same
transmission power $E$ does not make our analysis of outage
probability lose generality. This is because, in order to obtain a
general analysis of a cooperative network, it is sufficient to
make the average SNR of every channel different as shown in
\cite{ribeiro}. Although we make the transmission power at every
terminal the same, the variances $\Omega_h$, $\Omega_f$, and
$\Omega_g$ of the channels are different in general. As a result,
the average SNR of every channel is different, which makes our
analysis still general. Furthermore, the setting that every
terminal has the same transmission power $E$ does not affect the
diversity-multiplexing tradeoff analysis neither as shown in
\cite{azarian}, \cite{zheng}.} Consequently, the received signal
$y_{S_1}(2)$ of $\rmS_1$ at the second time slot is given by
\begin{eqnarray}
y_{S_1}(2)&=&\sqrt{E}\rho hy_R(1)+n_{S_1}(2)\\
&=&\rho h^2 Ex_1+ \rho hfEx_2 +\rho h\sqrt{E} n_R(1)+n_{S_1}(2),
\end{eqnarray}
where $n_{S_1}(2)$ is the additive Gaussian noise. In order to
ensure that the transmission power at $\rmR$ is always $E$, the
amplifying coefficient is chosen as
\begin{equation}
\rho = \sqrt{\frac{1}{E|h|^2+E|f|^2+1}}.
\end{equation}

Note that the received signal $y_{S_1}(2)$ contains both $x_1$ and
$x_2$, where only $x_2$ is the desired signal and $x_1$ is
actually an interference to $\rmS_1$. Since $\rmS_1$ perfectly
knows $x_1$, it can completely remove $x_1$ from $y_{S_1}(2)$ and
obtain a new signal $\tilde{y}_{S_1}(2)$ given by
\begin{equation}
\tilde{y}_{S_1}(2) = \rho hfEx_2 +\rho h\sqrt{E}
n_R(1)+n_{S_1}(2).
\end{equation}
Consequently, the instantaneous SNR $\gamma_1^{\ANC}$ at $\rmS_1$
is given by
\begin{equation}\label{eqn:snr1anc}
\gamma_1^{\ANC} = \frac{\rho^2|hf|^2E^2}{E\rho^2|h|^2+1}
\approx \frac{E|hf|^2}{2|h|^2+|f|^2},
\end{equation}
where the approximation is by letting $\rho\approx
\sqrt{1/(E|h|^2+E|f|^2)}$. Since this approximation is very tight
in the whole SNR range, it has been used in many previous
publications \cite{anghel1}, \cite{ribeiro}. Similarly, the
instantaneous SNR $\gamma_2^{\ANC}$ at $\rmS_2$ is given by
\begin{equation}\label{eqn:snr2anc}
\gamma_2^{\ANC} = \frac{\rho^2|hf|^2E^2}{E\rho^2|f|^2+1}
\approx \frac{E|hf|^2}{2|f|^2+|h|^2}.
\end{equation}

\subsection{Time Division Broadcast (TDBC)}

In order to accomplish information exchange between $\rmS_1$ and
$\rmS_2$, the TDBC protocol, as illustrated in Fig.\
\ref{fig:model}(b), was studied in \cite{kim1},
\cite{kim2}.\footnote{In \cite{shengli1} and
\cite{kim1}--\cite{liu}, the TDBC protocol was actually studied
when the relay worked in the decode-and-forward mode. However, it
is very simple to extend it to the case that the relay works in
the amplify-and-forward mode as shown in this paper.} As in the
ANC protocol, we still assume that every terminal has the same
transmission power $E$. At the first time slot, $\rmS_1$ transmits
$x_1$ to $\rmR$ and $\rmS_2$. Thus, the received signals $y_R(1)$
of $\rmR$ and $y_{S_2}(1)$ of $\rmS_2$ at the first time slot are
given by
\begin{eqnarray}
y_R(1)=\sqrt{E}hx_1+n_R(1),&& y_{S_2}(1)=\sqrt{E}gx_1+n_{S_2}(1),
\end{eqnarray}
where $n_R(1)$ and $n_{S_2}(1)$ are the additive Gaussian noises.
At the second time slot, $\rmS_2$ transmits $x_2$ to $\rmR$ and
$\rmS_1$. The received signals $y_R(2)$ of $\rmR$ and $y_{S_1}(2)$
of $\rmS_1$ at the second time slot are given by
\begin{eqnarray}
y_R(2)=\sqrt{E}fx_2+n_R(2),&& y_{S_1}(2)=\sqrt{E}gx_2+n_{S_1}(2),
\end{eqnarray}
where $n_R(2)$ and $n_{S_1}(2)$ are the additive Gaussian noises.
At the third time slot, $\rmR$ combines $y_R(1)$ and $y_R(2)$ at
first. Then it broadcasts the combined signal to $\rmS_1$ and
$\rmS_2$. The combined signal at $\rmR$ is denoted by $x_R$ and it
is given by
\begin{equation}
x_R = \eta_1y_R(1) + \eta_2y_R(2),
\end{equation}
where
\begin{eqnarray}
\eta_1 = \sqrt{\frac{\xi }{E|h|^2+1}},&&\eta_2 =
\sqrt{\frac{(1-\xi) }{E|f|^2+1}}.
\end{eqnarray}
The choice of $\eta_1$ and $\eta_2$ ensures that $x_R$ always has
unit power. The value of $\xi$ decides how the relay combines the
signals received from two different sources and it can be used to
optimize the performance of the TDBC protocol, which will be
discussed in detail later.

At the third time slot, the received signal $y_{S_1}(3)$ of
$\rmS_1$ is given by
\begin{eqnarray}
y_{S_1}(3) &=&
\sqrt{E}hx_R+n_{S_1}(3)\\
&=&\eta_1h^2Ex_1+\eta_2hfEx_2+\eta_1h\sqrt{E}n_R(1)+\eta_2h\sqrt{E}n_R(2)+n_{S_1}(3).
\end{eqnarray}
As in the ANC protocol, $\rmS_1$ can completely remove $x_1$ and
obtain a new signal $\tilde{y}_{S_1}(3)$ given by
\begin{equation}
\tilde{y}_{S_1}(3)=\eta_2hfEx_2+\eta_1h\sqrt{E}n_R(1)+\eta_2h\sqrt{E}n_R(2)+n_{S_1}(3).
\end{equation}
Then $\rmS_1$ combines $\tilde{y}_{S_1}(3)$ with $y_{S_1}(2)$ by
maximum ratio combining and the instantaneous SNR
$\gamma_1^{\TDBC}$ of the combined signal is
\begin{eqnarray}
\gamma_1^{\TDBC} =
E|g|^2+\frac{E^2\eta_2^2|hf|^2}{E|h|^2(\eta_1^2+\eta_2^2)+1}
\approx E|g|^2+\frac{(1-\xi)E|hf|^2}{|f|^2(\xi
+1)+(1-\xi)|h|^2},\label{eqn:snr1tdbc}
\end{eqnarray}
where the approximation is due to
$\eta_1\approx\sqrt{\xi/(E|h|^2)}$ and
$\eta_2\approx\sqrt{(1-\xi)/(E|f|^2)}$. Similarly, the
instantaneous SNR $\gamma_2^{\TDBC}$ at $\rmS_2$ is
\begin{eqnarray}
\gamma_2^{\TDBC} =
E|g|^2+\frac{E^2\eta_1^2|hf|^2}{E|f|^2(\eta_1^2+\eta_2^2)+1}
&\approx& E|g|^2+\frac{\xi E|hf|^2}{|h|^2(2-\xi)+\xi
|f|^2}.\label{eqn:snr2tdbc}
\end{eqnarray}

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

In this section, we present some numerical results to demonstrate
the performance of the ANC and TDBC protocols. We assume that all
three terminals are located in a straight line and $\rmR$ is
between $\rmS_1$ and $\rmS_2$. We fix the distance between
$\rmS_1$ and $\rmS_2$ as one and let $D_1$ denote the distance
between $\rmS_1$ and $\rmS_2$. Furthermore, we set the path loss
factor as four in order to model radio propagation in urban areas
\cite{rappaport}. As a result, the values of $\Omega_g$,
$\Omega_h$ and $\Omega_f$ equal to one, $D_1^{-4}$, and
$(1-D_1)^{-4}$, respectively.

In Fig.\ \ref{fig:tradeoff}, we compare the diversity-multiplexing
tradeoffs of the ANC and TDBC protocols. For each protocol, its
finite-SNR diversity-multiplexing tradeoff indeed converges to the
infinite-SNR case when $E$ goes to infinity. When $E$ is fixed,
the tradeoff curves of those two protocols have a cross point at
approximately $r=1/2$ as expected by Corollary
\ref{cor:crosspoint}. Fig.\ \ref{fig:tradeoff2} shows the values
of $Q(E)$, and hence, it demonstrates when the ANC and TDBC
protocols have the same diversity gain. One can see that our
approximate solution $\tilde{Q}(E)$ is extremely tight to the
exact one $Q(E)$ which is obtained by a numerical method.

In Fig.\ \ref{fig:outagedis}, we compare the exact outage
probabilities of the ANC and TDBC protocols with our lower bounds
given in (\ref{eqn:outANC}) and (\ref{eqn:outTDBC}). It can be
seen that the lower bounds are extremely tight for both protocols
even when we change the location of the relay and the value of
multiplexing gain. Furthermore, in Figs.\ \ref{fig:outagpower1}
and \ref{fig:outagpower2}, we see that our lower bounds are
constantly tight in the whole SNR range. In Fig.\
\ref{fig:outagpower1}, we let the multiplexing gain equal to zero.
As a result, the TDBC protocol achieves a higher diversity gain
than the ANC protocol, which coincides with our conclusion drawn
from Fig.\ \ref{fig:tradeoff}. On the other hand, we let the
multiplexing gain equal to $0.6$ in Fig.\ \ref{fig:outagpower2}
and show that the ANC protocol has a higher diversity gain for
this case.

In Figs.\ \ref{fig:outagpower1} and \ref{fig:outagpower2}, one can
also see that the optimum power allocation scheme of the ANC
protocol substantially reduces the outage probability even when
the relay is exactly in the middle of the two sources. For the
TDBC protocol, its outage probability is considerably reduced as
well when the proposed combing method given in (\ref{eqn:xitdbc1})
is implemented with equal power allocation. Moreover, by letting
$\xi$ equal to (\ref{eqn:xitdbc1}) and adopting equal power
allocation, we can achieve almost the same outage probability as
jointly optimizing the value of $\xi$ and the transmission powers
by numerical ways.

In Fig.\ \ref{fig:ratedis}, we show that the optimum power
allocation scheme can considerably increase the total mutual
information of the ANC protocol especially when $\rmR$ is close to
either $\rmS_1$ or $\rmS_2$. Moreover, the proposed optimum
combing method with equal power allocation can greatly increase
the total mutual information of the TDBC protocol as well, but the
improvement is not as considerable as that achieved by joint
optimization of $\xi$ and the transmission powers.

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

This paper studies the ANC and TDBC protocols which are used to
achieve information exchange in bidirectional cooperative
networks. We derive lower bounds of the outage probabilities of
those two protocols. The lower bounds are extremely tight in the
whole SNR range irrespective of the values of channel variances.
Therefore, based on those lower bounds, we derive the finite-SNR
diversity-multiplexing tradeoffs of the ANC and TDBC protocols.
Furthermore, we propose an optimum power allocation scheme for the
ANC protocol. This scheme can simultaneously minimize the outage
probability and maximize the total mutual information of the ANC
protocol. For the TDBC protocol, we develop an optimum combing
method for the relay terminal under an equal power allocation
assumption. This method substantially reduces the outage
probability and increases the total mutual information as well.

\section{Outage Probability and Finite-SNR Diversity-Multiplexing
Tradeoff}\label{sec:prob}

In this section, we derive lower bounds of the outage
probabilities of the ANC and TDBC protocols. They are very tight
in the whole SNR range, irrespective of the values of channel
variances. Furthermore, based on those lower bounds, we derive the
finite-SNR diversity-multiplexing tradeoffs of the ANC and TDBC
protocols.

\subsection{Analog Network Coding}\label{sec:ANCout}

When the ANC protocol is used, it follows from (\ref{eqn:snr1anc})
and (\ref{eqn:snr2anc}) that the mutual information at $\rmS_1$
and $\rmS_2$ is given by
\begin{eqnarray}\label{eqn:ancr1r2}
I_1^\ANC =
\frac{1}{2}\log\left(1+\gamma_1^{\ANC}\right),&&I_2^\ANC =
\frac{1}{2}\log\left(1+\gamma_2^{\ANC}\right).
\end{eqnarray}
Note that the pre-log factor $1/2$ is because the information
exchange between the two sources takes two time slots
\cite{rankov}. Assume that the target rate of the whole
bidirectional cooperative network is $R$. Since the two sources in
this network are equivalent terminals, it is fair to set the
target rate of each source as $R/2$. Furthermore, in a
bidirectional cooperative network, the two sources are not only
transmitters but also receivers. Due to this reason, a
bidirectional cooperative network can be seen as a multiuser
system. It is well known that a multiuser system is in outage when
any user is in outage \cite{tse2}, \cite{tse3}. Therefore, the ANC
protocol is in outage when either $I_1^{\ANC}$ or $I_2^\ANC$ is
smaller than the target rate $R/2$, i.e.
\begin{equation}\label{eqn:ancout}
P_{\ANC}^{\outage}(R) = \Pr\left(I_1^\ANC<\frac{R}{2} \ \
{\mathrm{or}} \ \ I_2^\ANC<\frac{R}{2}\right).
\end{equation}

The exact expression of the outage probability is very hard to
derive even after we approximate the instantaneous SNRs as in
(\ref{eqn:snr1anc}) and (\ref{eqn:snr2anc}). However, it is well
known that the harmonic mean of two positive numbers can be
upper-bounded by the minimum of those two numbers as follows
\cite{anghel1}:
\begin{equation}\label{eqn:hormonic}
\frac{xy}{x+y}<\min(x,y).
\end{equation}
In order to make the analysis feasible, we use this method to
upper-bound the instantaneous SNRs and derive a lower bound of the
outage probability in the following lemma.\footnote{In fact, the
harmonic mean can also be lower-bounded as $1/2\min(x,y)\leq
xy/(x+y)$ \cite{anghel1}. Using this fact and the techniques
developed in Appendix A, we can find an upper bound of the outage
probability as well. However, this upper bound is not as tight as
the lower bound given in Lemma \ref{thm:ancoutage}, and hence, it
is not presented in this paper.}

\begin{lemma}\label{thm:ancoutage}
The outage probability of the ANC protocol can be lower-bounded as
follows:
\begin{eqnarray}\label{eqn:outANC}
P_{\ANC}^{\outage}(R)>\tilde{P}_{\ANC}^{\outage}(R)= 1- \exp\left(
-\frac{2\left(\Omega_h+\Omega_f\right)\left(2^{R}-1\right)}{E\Omega_h\Omega_f}\right).
\end{eqnarray}
\end{lemma}
\begin{proof}
See Appendix A.
\end{proof}

We notice that, irrespective of the values of channel variances,
the lower bound $\tilde{P}_\ANC^\outage(R)$ is extremely tight in
the whole SNR range as shown in Figs.\
\ref{fig:outagedis}--\ref{fig:outagpower2}. Due to this reason, we
use this lower bound to find the finite-SNR diversity-multiplexing
tradeoff of the ANC protocol as in \cite{nara} and it is given in
the follow theorem.
\begin{theorem}\label{cor:anccurve}
The finite-SNR diversity-multiplexing tradeoff curve $d^\ANC(r,E)$
of the ANC protocol is given by
\begin{eqnarray}\label{eqn:anccurvefinite}
d^\ANC(r,E)=\frac{2(\Omega_h+\Omega_f)\left(1+rE(1+E)^{r-1}-(1+E)^r\right)}
{E\Omega_h\Omega_f\left(1-\exp\left(\frac{2(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f}\left((1+E)^r-1\right)\right)\right)}
,&&0\leq r\leq 1.
\end{eqnarray}
\end{theorem}
\begin{proof}
By definition, the finite-SNR diversity-multiplexing tradeoff
curve $d^\ANC(r,E)$ is given by \cite{nara}
\begin{equation}\label{eqn:anctradeoffdef}
d^\ANC(r,E) = -\frac{\partial \ln
\tilde{P}^\outage_\ANC(r\log(1+aE))}{\partial \ln E},
\end{equation}
where $a$ is a constant and called array gain. Since every
terminal has only one antenna, we set $a=1$ according to its
definition in \cite{nara}. By substituting (\ref{eqn:outANC}) into
(\ref{eqn:anctradeoffdef}), one can easily obtain
(\ref{eqn:anccurvefinite}).
\end{proof}

An interesting special case of the trade-off curve $d^\ANC(r,E)$
is when the SNR goes to infinity. This actually corresponds to the
definition of infinite-SNR diversity-multiplexing tradeoff given
in \cite{zheng}. For this case, the tradeoff curve of the ANC
protocol becomes
\begin{equation}\label{eqn:anccurveinfinite}
\lim_{E\rightarrow\infty}d^\ANC(r,E) = 1-r, \ \ \ \ 0\leq r\leq 1.
\end{equation}
Based on (\ref{eqn:anccurveinfinite}), we see that the maximum
multiplexing gain of the ANC protocol is one. This is because,
although each traffic flow in the ANC protocol takes two time
slots to complete the transmission, there are two concurrent
traffic flows and they are supported by the same physical
channels. \footnote{Actually, we conjecture that the highest
multiplexing gain any bidirectional cooperative network can
achieve is also one. It is certainly necessary to find the optimum
diversity-multiplexing tradeoff of a bidirectional cooperative
network as the authors did for the unidirectional cooperative
networks in \cite{azarian}, \cite{yang}. However, this is beyond
the scope of this paper where we focus on deriving and comparing
the tradeoffs of two specific protocols.} Note that, in a
conventional unidirectional cooperative network, there is only one
traffic flow in the network, and hence, the maximum multiplexing
gain of such network is just $1/2$. As a result, the ANC protocol
indeed has much higher bandwidth efficiency than the conventional
unidirectional cooperative networks. However, the maximum
diversity gain of the ANC protocol is just one. This is because
the ANC protocol let the two sources transmit simultaneously at
the first time slot. As a result, the sources can not receive the
signals from the direct channel due to the half-duplex constraint.
In order to achieve higher diversity gain, we can assign one time
slot to each source and let them transmit separately as in the
TDBC protocol discussed in the next subsection. As we will show,
however, this leads to some loss of multiplexing gain.

\subsection{Time Division Broadcasting}

When a bidirectional cooperative network implements the TDBC
protocol, the mutual information at $\rmS_1$ and $\rmS_2$ is given
by
\begin{eqnarray}\label{eqn:tdbci1i2}
I_1^\TDBC =
\frac{1}{3}\log\left(1+\gamma_1^\TDBC\right),&&I_2^\TDBC =
\frac{1}{3}\log\left(1+\gamma_2^\TDBC\right),
\end{eqnarray}
where the instantaneous SNRs are given by (\ref{eqn:snr1tdbc}) and
(\ref{eqn:snr2tdbc}), respectively. The pre-log factor $1/3$ is
because each traffic flow takes three time slots in the TDBC
protocol. As for the ANC protocol, the TDBC protocol is in outage
when either $I_1^\TDBC$ or $I_2^\TDBC$ is smaller than the target
rate $R/2$, i.e.
\begin{equation}\label{eqn:tdbcout}
P_{\TDBC}^{\outage}(R) = \Pr\left(I_1^\TDBC<\frac{R}{2} \ \
{\mathrm{or}} \ \ I_2^\TDBC<\frac{R}{2}\right).
\end{equation}
It is very hard to obtain the exact expression of
$P_{\TDBC}^{\outage}(R)$ even after using the approximations in
(\ref{eqn:snr1tdbc}) and (\ref{eqn:snr2tdbc}). In order to make
the problem tractable, we still use (\ref{eqn:hormonic}) to
upper-bound the instantaneous SNRs.
Furthermore, we let $\xi=1/2$ to simplify the derivation in this
subsection. This actually means that $\rmR$ uses half of its power
to transmit to $\rmS_1$ and uses the other half to $\rmS_2$. In
Subsection \ref{sec:TDBCpower}, we will consider how $\rmR$ can
optimally allocate its transmission power for $\rmS_1$ and
$\rmS_2$ by choosing an optimum value of $\xi$. A lower bound of
the outage probability of the TDBC protocol is given in the
following lemma.
\begin{lemma}\label{thm:outTDBC}
When $\xi=1/2$, the outage probability $P_\TDBC^\outage(R)$ of the
TDBC protocol can be lower-bounded as follows:
\begin{eqnarray}\label{eqn:outTDBC}
P_\TDBC^\outage(R)>\tilde{P}_\TDBC^\outage(R)&=&
1-\frac{1}{3\Omega_f\Omega_g+3\Omega_h\Omega_g-\Omega_h\Omega_f}\nonumber\\
&&\times\left(
3\Omega_g\left(\Omega_f+\Omega_h\right)\exp\left(-\frac{2^{1.5R}-1}{E\Omega_g}\right)\right.\nonumber\\
&&\left.-\Omega_h\Omega_f\exp\left(-\frac{3(2^{1.5R}-1)(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f}\right)
\right).
\end{eqnarray}
\end{lemma}
\begin{proof}
See Appendix B.
\end{proof}

Figs.\ \ref{fig:outagedis}--\ref{fig:outagpower2} demonstrate that
this lower bound $\tilde{P}_\TDBC^\outage(R)$ is extremely tight
in the whole SNR range, irrespective of the values of channel
variances. Therefore, we use this lower bound to find the
finite-SNR diversity-multiplexing tradeoff of the TDBC protocol
and it is given in the follow theorem.
\begin{theorem}\label{cor:tdbccurve}
The finite-SNR diversity-multiplexing tradeoff curve
$d^\TDBC(r,E)$ of the TDBC protocol is given by
\begin{eqnarray}
d^\TDBC(r,E) = \frac{d_1(r,E)}{d_2(r,E)},&& 0\leq r\leq
\frac{2}{3},
\end{eqnarray}
where
\begin{eqnarray}
d_1(r,E) &=&
3(\Omega_h+\Omega_f)(\lambda(3rE-2E-2)+2E+2)\nonumber\\
&&\times\left(\exp\left(-\frac{3(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f}(\lambda-1)\right)
-\exp\left(-\frac{\lambda-1}{E\Omega_g}\right)
\right),\\
d_2(r,E)&=&2E(E+1)\left(3\Omega_g(\Omega_h+\Omega_f)\left( 1-
\exp\left(-\frac{\lambda-1}{E\Omega_g}\right)\right)\right.\nonumber\\
&&\left.-\Omega_h\Omega_f\left(1-
\exp\left(-\frac{3(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f}(\lambda-1)\right)\right)\right),\\
\lambda&=&(1+E)^{\frac{3r}{2}}.
\end{eqnarray}
\end{theorem}
\begin{proof}
The proof is essentially the same as that of Theorem
\ref{cor:anccurve}, except that we use the lower bound
$\tilde{P}_\TDBC^\outage(R)$ of the TDBC protocol instead of
$\tilde{P}_\ANC^\outage(R)$.
\end{proof}

When the SNR goes to infinity, the trade-off curve $d^\TDBC(r,E)$
becomes
\begin{eqnarray}\label{eqn:tdbccurveinfinite}
\lim_{E\rightarrow\infty} d^\TDBC(r,E) = 2 - 3r, &&0\leq r\leq
\frac{2}{3}.
\end{eqnarray}
As a result, the maximum multiplexing gain of the TDBC protocol is
$2/3$, which is smaller than that of the ANC protocol. This is
because, although the TDBC protocol supports two traffic flows
concurrently as the ANC protocol, each traffic flow takes three
time slots to complete the transmission as opposed to two time
slots in the ANC protocol. Due to this reason, the bandwidth
efficiency of the TDBC protocol is not as high as that of the ANC
protocol. However, note that the maximum multiplexing gain of the
TDBC protocol is still larger than that of the conventional
unidirectional cooperative networks. On the other hand, the
maximum diversity gain of the TDBC protocol is indeed two as shown
in \cite{kim1}, \cite{kim2}, and it is much larger than that of
the ANC protocol. Moreover, based on (\ref{eqn:anccurveinfinite})
and (\ref{eqn:tdbccurveinfinite}), those two protocols achieve the
same diversity gain at $r=1/2$ when SNR goes to infinity.

Many previous publications on bidirectional cooperative networks
did not consider the direct channel between the two sources
\cite{shengli1}, \cite{katti2}, \cite{shengli2}, and
\cite{liu}--\cite{lee2}. As a result, the maximum diversity gain
of the TDBC protocol is just one which is the same as that of the
ANC protocol. Since the maximum multiplexing gain of the TDBC
protocol is never larger than that of the ANC protocol, the ANC
protocol always outperforms the TDBC protocol in terms of
diversity-multiplexing tradeoff. When the direct channel exists,
however, the ANC protocol no longer outperforms the TDBC protocol
for all the time. As shown in Fig.\ \ref{fig:tradeoff}, those two
protocols achieve the same diversity gain when the multiplexing
gain is approximately $1/2$. When the multiplexing gain becomes
smaller, the TDBC protocol outperforms the ANC protocol as it
achieves a higher diversity gain; while, when the multiplexing
gain becomes bigger, the ANC protocol outperforms. This implies
that the ANC protocol can transmit information more efficiently,
but the TDBC protocol can transmit information more reliably.
Therefore, one may alternatively use those two protocols depending
on the specific task of a bidirectional cooperative network.

Certainly, it is desirable to find when the ANC and TDBC protocols
have the same diversity gain for a fixed $E$ by solving the
equation $d^\ANC(r,E)=d^\TDBC(r,E)$. Let $Q(E)$ denote the
solution to this equation. However, the exact expression of $Q(E)$
can not be given in closed form. In the following corollary, we
present a very accurate approximation to $Q(E)$.

\begin{cor}\label{cor:crosspoint}
When $E$ is fixed, the solution $Q(E)$ to the equation
$d^\ANC(r,E)=d^\TDBC(r,E)$ can be approximated by
\begin{eqnarray}
Q(E)\approx\tilde{Q}(E) =
\frac{1}{2}-\frac{\nu_1-\mu_1}{\nu_2-\mu_2}.
\end{eqnarray}
The coefficients $\mu_1$, $\mu_2$, $\nu_1$, and $\nu_2$ are given
as follows:
\begin{eqnarray}
\mu_1 &=& \frac{2\left( \Omega_h+\Omega_f \right)  \left( \,{\frac
{E}{2\sqrt
{1+E}}}+1-\sqrt {1+E} \right)}{ {E}{\Omega_h}{\Omega_f} \left( 1-c_1 \right)},\label{eqn:mu1}\\
\mu_2 &=&
\frac{2(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f(1-c_1)}\left(\frac{E\ln(1+E)+2E}{2\sqrt{1+E}}-\sqrt{1+E}\ln(1+E)\right.\\
&&\left.+\frac{c_1(\Omega_h+\Omega_f)(E-2\sqrt{1+E}+2)\ln(1+E)}{E\Omega_h\Omega_f(c_1-1)}
\right),\label{eqn:mu2}\\
\nu_1&=&\frac{3(c_3-c_2)(\Omega_h+\Omega_f)\left(4(1+E)^{\frac{1}{4}}-E-4\right)}{E(1+E)^{\frac{1}{4}}
(3\Omega_g(\Omega_h+\Omega_f)(1-c_2)-\Omega_h\Omega_f(1-c_3))},\label{eqn:nu1}
\end{eqnarray}
\begin{eqnarray}
\nu_2 &=& \frac{3(\Omega_h+\Omega_f)}{E(1+E)^{\frac{1}{4}}}\left((c_3-c_2)\left(E-\frac{1}{4}(E+4)\ln(1+E)\right)\right.\nonumber\\
&&\left.+\frac{\ln(1+E)}{2E}\left(\frac{c_2}{\Omega_g}
-\frac{3c_3(\Omega_h+\Omega_f)}{\Omega_h\Omega_f}\right)(1+E)^{\frac{3}{4}}\left(2(1+E)^{\frac{1}{4}}
-\frac{E}{2}-2 \right) \right)\nonumber\\
&&+\frac{9(c_2-c_3)^2(\Omega_h+\Omega_f)^2\left(E+4-4(1+E)^{\frac{1}{4}}\right)(1+E)^{\frac{3}{4}}\ln(1+E)}
{4E^2(1+E)^{\frac{1}{4}}(3\Omega_g(\Omega_h+\Omega_f)(c_2-1)+\Omega_h\Omega_f(1-c_3))},\label{eqn:nu2}
\end{eqnarray}
where $c_1$, $c_2$, $c_3$ are constants and they are given by
\begin{eqnarray}
c_1 &=&\exp\left(\frac { 2\left( \Omega_h+\Omega_f \right) \left(
\sqrt {1+E}-1 \right) }{E\Omega_h\Omega_f}\right)\\
c_2 &=&\exp\left(\frac{1-(1+E)^{\frac{3}{4}}}{E\Omega_g}\right)\\
c_3
&=&\exp\left(\frac{3(\Omega_h+\Omega_f)(1-(1+E)^{\frac{3}{4}})}{E\Omega_g}\right).
\end{eqnarray}
\end{cor}
\begin{proof}
In Fig.\ \ref{fig:tradeoff}, we notice that both $d^\ANC(r,E)$ and
$d^\TDBC(r,E)$ can be accurately approximated by linear functions.
Furthermore, the solution $Q(E)$ to this equation
$d^\ANC(r,E)=d^\TDBC(r,E)$ is very close to $r=1/2$, irrespective
of the value of $E$. Therefore, we approximate $d^\ANC(r,E)$ and
$d^\TDBC(r,E)$ by Taylor expansion as follows:
\begin{eqnarray}\label{eqn:taylor}
d^\ANC(r,E) \approx \mu_1+\mu_2\left(r-\frac{1}{2}\right),&&
d^\TDBC(r,E) \approx \nu_1+\nu_2\left(r-\frac{1}{2}\right),
\end{eqnarray}
where the coefficients $\mu_1$, $\mu_2$, $\nu_1$, and $\nu_2$ are
given in (\ref{eqn:mu1})--(\ref{eqn:nu2}). By using
(\ref{eqn:taylor}), one can easily obtain $\tilde{Q}(E)$.
\end{proof}

Fig.\ \ref{fig:tradeoff2} demonstrates that $\tilde{Q}(E)$ is a
very tight approximation of $Q(E)$. When the SNR goes to infinity,
we can show that
\begin{equation}
\lim_{E\rightarrow\infty}\tilde{Q}(E) = \frac{1}{2}.
\end{equation}
This coincides with our conclusion based on
(\ref{eqn:anccurveinfinite}) and (\ref{eqn:tdbccurveinfinite}).
Furthermore, it is not hard to analytically show that
$\tilde{Q}(E)$ is always smaller than or equal to $1/2$ for any
$E$, which implies that the ANC protocol achieves a higher
diversity gain than the TDBC protocol as long as the multiplexing
gain is lager than $1/2$.

Note that our outage probability lower bounds and finite-SNR
diversity-multiplexing tradeoffs of the ANC and TDBC protocols are
based on the assumption that there is a direct channel between the
two sources. When such direct channel does not exist as in
\cite{shengli1}, \cite{katti2}, \cite{shengli2}, and
\cite{liu}--\cite{haoy}, however, out results can be easily
extended to this special case by letting $\Omega_g=0$. For the ANC
protocol, the lower bound $\tilde{P}_\ANC^\outage(R)$ and
diversity-multiplexing tradeoff $d^\ANC(r,E)$ do not change. For
the TDBC protocol, the lower bound of the outage probability and
the diversity-multiplexing tradeoff now become
\begin{eqnarray}
\tilde{P}_\TDBC^\outage(R)&=&
1-\exp\left(-\frac{3(2^{1.5R}-1)(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f}
\right),
\end{eqnarray}
\begin{equation}
d^\TDBC(r,E)=\frac{3(\Omega_h+\Omega_f)(\lambda(3rE-2E-2)+2E+2)
\exp\left(-\frac{3(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f}(\lambda-1)\right)
}{2E(E+1)\Omega_h\Omega_f\left(
\exp\left(-\frac{3(\Omega_h+\Omega_f)}{E\Omega_h\Omega_f}(\lambda-1)\right)-1\right)}.
\end{equation}
Furthermore, when the SNR goes to infinity, the trade-off curve of
the TDBC protocol becomes
\begin{eqnarray}\label{eqn:tdbccurveinfinite2}
\lim_{E\rightarrow\infty} d^\TDBC(r,E) = 1 - \frac{3}{2}r, &&0\leq
r\leq \frac{2}{3}.
\end{eqnarray}
By comparing (\ref{eqn:tdbccurveinfinite2}) and
(\ref{eqn:anccurveinfinite}), one can see that the ANC protocol
indeed always outperforms the TDBC protocol in terms of
diversity-multiplexing tradeoff when the direct channel does not
exist. This coincides with the conclusion which can be drawn from
\cite{shengli1}, \cite{katti2}, and \cite{haoy}.

\section{Optimum Power Allocation}\label{sec:power}

In this section, we first propose an optimum power allocation
scheme for the ANC protocol. This scheme can simultaneously
minimize the outage probability and maximize the total mutual
information of the ANC protocol. Secondly, we develop an optimum
method for the TDBC protocol to combine the received signals at
the relay. This method also minimizes the outage probability and
maximizes the total mutual information of the TDBC protocol at the
same time.

\subsection{Analog Network Coding}

In Subsection \ref{sec:ANCout}, we derive a lower bound of the
outage probability of the ANC protocol when every terminal has the
same transmission power $E$. Since every terminal knows the values
of $h$ and $f$, it is more desirable to allocate the transmission
power according to channel conditions in order to maximize the
performance. Such power allocation problem has not been
investigated yet in previous publications. We now assume that the
transmission powers of $\rmS_1$, $\rmS_2$, and $\rmR$ are $E_1$,
$E_2$, and $E_r$, respectively. Consequently, the instantaneous
SNRs at $\rmS_1$ and $\rmS_2$ should be rewritten as
\begin{eqnarray}\label{eqn:ancsnr12gen}
\gamma_1^{\ANC}
\approx\frac{E_rE_2|hf|^2}{(E_r+E_1)|h|^2+E_2|f|^2},&&
\gamma_2^{\ANC} \approx
\frac{E_rE_1|hf|^2}{(E_r+E_2)|f|^2+E_1|h|^2}.
\end{eqnarray}

When it comes to optimum power allocation, two optimization goals
are usually considered: minimization of the outage probability and
maximization of the total mutual information.\footnote{For a
single user system whose outage probability is formulated by
$\Pr(I<R)$, it is not hard to see that the optimum power
allocation scheme that minimizes the outage probability also
maximizes the mutual information. For the bidirectional
cooperative network considered in this paper, however, its outage
probability is given in the form $\Pr(I_1<R/2\ {\mathrm{or}}\
I_2<R/2)$ and its total mutual information is $I_1+I_2$.
Therefore, it is not easy to see if there exists an optimum power
allocation scheme which can minimize the outage probability and
maximize the total mutual information at the same time.} We first
use the outage probability as the metric to optimally allocate the
power. That is, the optimization problem is formulated by
\begin{eqnarray}\label{eqn:ancopt1}
(E_1,E_2,E_r) = \arg\min_{E_1,E_2,E_r} P_\ANC^\outage(R),
&&{\mathrm{subject \ to}}\ \ \ E_1+E_2+E_r=3E.
\end{eqnarray}
It follows from (\ref{eqn:ancout}) that the optimization problem
in (\ref{eqn:ancopt1}) is equivalent to the following one
\begin{eqnarray}\label{eqn:ancminmax}
(E_1,E_2,E_r) = \arg\max_{E_1,E_2,E_r}\min
(\gamma_1^\ANC,\gamma_2^\ANC),&&{\mathrm{subject \ to}}\ \ \
E_1+E_2+E_r=3E.
\end{eqnarray}
The minimax problem in (\ref{eqn:ancminmax}) is solved in the
following theorem.
\begin{theorem}\label{thm:ancpower1}
When $E_1+E_2+E_r=3E$, the optimum power allocation that minimizes
the outage probability $P_\ANC^\outage(R)$ of the ANC protocol is
given by
\begin{eqnarray}
E_r &=& \frac{3}{2}E,\label{eqn:ancer}\\
E_1 &=& \frac{3|f|}{2(|h|+|f|)}E,\label{eqn:ance1}\\
E_2 &=& \frac{3|h|}{2(|h|+|f|)}E\label{eqn:ance2}.
\end{eqnarray}
\end{theorem}
\begin{proof}
See Appendix C.
\end{proof}

Secondly, we consider the optimum power allocation scheme that
maximizes the total mutual information $I^\ANC=I_1^\ANC+I_2^\ANC$
of the ANC protocol. Therefore, the optimization problem is
formulated as
\begin{eqnarray}\label{eqn:ancoptth1}
(E_1,E_2,E_r) = \arg\max_{E_1,E_2,E_r} I^\ANC, &&{\mathrm{subject
\ to}}\ \ \ E_1+E_2+E_r=3E.
\end{eqnarray}
This optimization problem is solved in the following lemma.
\begin{lemma}
When $E_1+E_2+E_r=3E$, the optimum power allocation scheme that
maximizes the total mutual information $I^\ANC$ of the ANC
protocol is given by (\ref{eqn:ancer})--(\ref{eqn:ance2}).
\end{lemma}
\begin{proof}
Let $E_1=3\alpha E$, $E_2=3\beta E$, and $E_r =
3(1-\alpha-\beta)E$, where $0\leq \alpha\leq 1$, $0\leq \beta\leq
1$, and $\alpha+\beta\leq 1$. With those constraints, the solution
to $\partial I^\ANC/\partial \alpha=0$ and $\partial
I^\ANC/\partial\beta=0$ is given by
\begin{eqnarray}
\alpha = \frac{|f|}{2(|h|+|f|)},&& \beta = \frac{|h|}{2(|h|+|f|)}.
\end{eqnarray}
As a result, the optimum power allocation scheme is given by
(\ref{eqn:ancer})--(\ref{eqn:ance2}).
\end{proof}

Interestingly, the optimum power allocation schemes based on
outage probability and total mutual information are exactly the
same. This is due to the special structures of the instantaneous
SNRs $\gamma_1^\ANC$ and $\gamma_2^\ANC$ given in
(\ref{eqn:ancsnr12gen}). As a result, we obtain an optimum power
allocation scheme that simultaneously minimizes the outage
probability and maximizes the total mutual information of the ANC
protocol.

\subsection{Time Division Broadcasting}\label{sec:TDBCpower}

Unlike the ANC protocol, it is very hard to find an optimum power
allocation scheme for the TDBC protocol no matter using the outage
probability or the total mutual information as the criteria. For
example, if we intend to minimize the outage probability, the
optimization problem is a generalized fractional programming
problem. Such problem can only be solved numerically for most
cases \cite{barros}, \cite{crouzeix}.\footnote{In fact, for the
ANC protocol, the optimization problem (\ref{eqn:ancminmax}) is
also a generalized fractional programming problem. We obtain a
closed form solution to (\ref{eqn:ancminmax}) only because
$\gamma_1^\ANC$ and $\gamma_2^\ANC$ have very special structures.}
However, the optimum value of $\xi$ can be analytically found in
closed form and it can greatly improve the performance. We first
try to find the optimum value of $\xi$ that minimizes the outage
probability of the TDBC protocol. That is, we consider the
following optimization problem
\begin{eqnarray}\label{eqn:tdbcminmax1}
\xi = \arg\min_{\xi}P_\TDBC^\outage(R),&&{\mathrm{subject \ to}} \ \ \ 0\leq\xi\leq 1
\end{eqnarray}
It follows from (\ref{eqn:tdbcout}) that the optimization problem
in (\ref{eqn:tdbcminmax1}) is equivalent to the following one
\begin{eqnarray}\label{eqn:tdbcminmax2}
\xi = \arg\mathop{\max}_{\xi}\min
(\gamma_1^\TDBC,\gamma_2^\TDBC),&&{\mathrm{subject \ to}} \ \ \ 0\leq\xi\leq 1
\end{eqnarray}
The solution to (\ref{eqn:tdbcminmax2}) is given in a simple and
closed form in the following theorem.
\begin{theorem}\label{thm:tdbcout}
The optimum value of $\xi$ that minimizes the outage probability
$P_\TDBC^\outage(R)$ of the TDBC protocol is given by
\begin{equation}
\xi = \frac{|h|}{|h|+|f|}\label{eqn:xitdbc1}.
\end{equation}
\end{theorem}
\begin{proof}
See Appendix D.
\end{proof}

Note that the optimum value of $\xi$ given in Theorem
\ref{thm:tdbcout} is based on the assumption that every terminal
has the same transmission power $E$. Certainly, one can jointly
optimize the value of $\xi$ and the transmission power of every
terminal to minimize the outage probability, but this can only be
done numerically. Actually, by letting every terminal has the same
transmission power $E$ and letting $\xi$ equal to $|h|/(|h|+|f|)$
as in (\ref{eqn:xitdbc1}), the performance of the network is
almost the same as that of the network where the transmission
powers and the value of $\xi$ are jointly optimized by numerical
ways as shown in Figs.\ \ref{fig:outagpower1} and
\ref{fig:outagpower2}.

Secondly, we investigate the optimum value of $\xi$ that maximizes
the total mutual information $I^\TDBC=I_1^\TDBC+I_2^\TDBC$ of the
TDBC protocol, i.e.
\begin{eqnarray}\label{eqn:tdbcopt2}
\xi = \arg\max_{\xi} I^\TDBC,&& {\mathrm{subject \ to}} \ \ 0\leq
\xi\leq 1,
\end{eqnarray}
where $I_1^\TDBC$ and $I_2^\TDBC$ are given by
(\ref{eqn:tdbci1i2}). The solution to (\ref{eqn:tdbcopt2}) is
given the following lemma.
\begin{lemma}\label{lem:tdbcpower2}
The optimum value of $\xi$ that maximizes the total mutual
information $I^\TDBC$ of the TDBC protocol is given by
(\ref{eqn:xitdbc1}).
\end{lemma}
\begin{proof}
By solving the equation $\partial I^\TDBC/\partial \xi=0$, we can
easily obtained the solution given in (\ref{eqn:xitdbc1}).
\end{proof}

As for Theorem \ref{thm:tdbcout}, the optimum value of $\xi$ in
Lemma \ref{lem:tdbcpower2} is also based on the setting that every
terminal has the same transmission power $E$. The joint
optimization of the transmission powers and the value of $\xi$ to
maximize the total mutual information can only be accomplished by
numerical ways. In fact, by simply letting every terminal has the
same transmission power $E$ and letting $\xi$ equal to
(\ref{eqn:xitdbc1}), the total mutual information of the TDBC
protocol can be improved substantially as shown in Fig.\
\ref{fig:ratedis}.

Interestingly, we notice that the optimum values of $\xi$ in
Theorem \ref{thm:tdbcout} and Lemma \ref{lem:tdbcpower2} are
exactly the same, although they are based on two different
criteria. As a result, by letting $\xi$ equal to $|h|/(|h|+|f|)$,
we can minimize the outage probability and maximize the total
mutual information of the TDBC protocol at the same time.
Actually, the reason why we can find such $\xi$ is because every
terminal has the same transmission power. In general, when every
terminal has unequal transmission power, the solutions to
(\ref{eqn:tdbcminmax2}) and (\ref{eqn:tdbcopt2}) are different.
Furthermore, the joint optimization of the transmission powers and
the value of $\xi$ based on the outage probability criteria is
also generally different with that based on the total mutual
information criteria. Thus, one may use equal power allocation and
set $\xi$ as (\ref{eqn:xitdbc1}) in order to transmit information
efficiently and reliably at the same time.


text:
\section{Isotropic coordinates}

The interval of the static, spherically symmetric, gravitational field in vacuum, expressed in isotropic coordinates, was found by Weyl \cite{Weyl}:
\begin{equation}
ds^2=\frac{(1-r_g/(4r))^2}{(1+r_g/(4r))^2}c^2 dt^2-(1+r_g/(4r))^4(dr^2+r^2 d\Omega^2),
\label{interv1}
\end{equation}
where $0\le r<\infty$ is the radial coordinate, $d\Omega$ is the element of the solid angle, and $r_g=2GM/c^2$ is the Schwarzschild radius.
This metric does not change its form under the coordinate transformation:
\begin{equation}
r\rightarrow r'=\frac{r_g^2}{16r},
\label{transf}
\end{equation}
and is Galilean for $r\rightarrow\infty$.
Therefore it is also Galilean for $r\rightarrow 0$, describing an Einstein-Rosen bridge (wormhole): two Schwarzschild solutions to the Einstein field equations (a black hole and white hole) connected at the singular ($\textrm{det}\,g_{\mu\nu}=0$) surface $r=r_g/4$ (common event horizon) \cite{ER,FW}.
The nonzero components of the Riemann curvature tensor for this metric are given by
\begin{eqnarray}
& & R^{0\theta}_{\phantom{0\theta}0\theta}=R^{0\phi}_{\phantom{0\phi}0\phi}=R^{r\theta}_{\phantom{r\theta}r\theta}=R^{r\phi}_{\phantom{r\phi}r\phi}=\frac{r_g}{2r^3(1+r_g/(4r))^6}, \nonumber \\
& & R^{0r}_{\phantom{0r}0r}=R^{\theta\phi}_{\phantom{\theta\phi}\theta\phi}=-\frac{r_g}{r^3(1+r_g/(4r))^6},
\label{curv}
\end{eqnarray}
so the Kretschmann scalar is finite everywhere: $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=12r_g^2 r^{-6}(1+r_g/(4r))^{-12}$, going to zero as $r\rightarrow\infty$ and $r\rightarrow 0$.
The Einstein-Rosen metric for $r>r_g/4$ describes the exterior sheet of a Schwarzschild black hole (the transformation of the radial coordinate $r\rightarrow r_S=r(1+r_g/(4r))^2$ brings the interval (\ref{interv1}) into the standard Schwarzschild form \cite{Schw}).
The spacetime given by the metric (\ref{interv1}) for $r<r_g/4$ is regarded by an observer at $r>r_g/4$ as the interior of a black hole.
Because of the invariance of the metric (\ref{interv1}) under the transformation (\ref{transf}), this interior is an image of the other exterior sheet.
This situation is analogous to the method of image charges for spheres in electrostatics, where the interaction between an electric charge situated at a distance $r$ from the center of a conducting sphere of radius $R<r$ is equivalent to the interaction of the same charge with a charge of the opposite sign situated inside this sphere at a distance $R^2/r$ from its center \cite{LL8}.
The radius $R$ corresponds to the Schwarzschild surface in isotropic coordinates, $r=r_g/4$.

\section{Radial motion}

Consider a massive particle moving radially in the gravitational field described by the metric (\ref{interv1}).
For brevity, we use
\begin{equation}
h=g_{00}=\frac{(1-r_g/(4r))^2}{(1+r_g/(4r))^2},\,\,\,\,f=-g_{rr}=(1+r_g/(4r))^4.
\label{not}
\end{equation}
The motion of the particle is given by the radial geodesic equations.
If the particle is at rest at $r=r_0$, then these equations are
\begin{eqnarray}
& & \frac{dt}{d\tau}=u^0=\sqrt{h_0}/h, \label{mot1} \\
& & \frac{dr}{cd\tau}=u^r=\epsilon(h_0 h^{-1} f^{-1}-f^{-1})^{1/2},
\label{mot2}
\end{eqnarray}
where $\tau$ is the proper time of the particle, $h_0=h|_{r=r_0}$, and $\epsilon=-1\,(+1)$ for an infalling (outgoing) motion.
Consider a particle falling into a black hole, $\epsilon=-1$.
As $r\rightarrow r_g/4$, $h$ goes to zero and $f\rightarrow 16$, so both $u^0$ and $u^r$ become infinite.
Even if the initial motion were not purely radial, the components $u^0,\,u^r$ would still become infinite at $r=r_g/4$, with $u^\theta,\,u^\phi$ remaining finite.
Therefore, each motion of a massive particle becomes effectively radial at the surface $r=r_g/4$.

A distant observer situated in a nearly Galilean spacetime measures the velocity of the infalling particle as
\begin{equation}
v_d=\frac{dr}{dt}=c\frac{u^r}{u^0}=-c\frac{(h_0 f^{-1}h-f^{-1}h^2)^{1/2}}{\sqrt{h_0}}.
\label{mot3}
\end{equation}
As $r\rightarrow r_g/4$, $v_d$ goes to zero.
Writing $r=r_g/4+\xi$, where $0<\xi\ll r_g$, gives $v_d\approx -c\xi/(2r_g)$ and thus $r-r_g/4\sim\mbox{exp}(-ct/(2r_g))$, so the particle reaches the surface of a black hole $r=r_g/4$ after an infinite time $t$.
This surface is an event horizon for a distant observer, as it is for the standard Schwarzschild metric \cite{Lem}.
The proper time $\Delta\tau$ of the particle for moving radially from $r=r_0$ to $r=r_g/4$ is finite, which can be shown by considering $r_0=r_g/4+\xi$:
\begin{equation}
c\Delta\tau=\int_{r_g/4}^{r_g/4+\xi}dr/u^r\approx r_g.
\label{mot4}
\end{equation}
After reaching the surface $r=r_g/4$, the particle continues moving; its radial coordinate $r$ decreases to $r_1=r_g^2/(16r_0)\approx r_g/4-\xi$ (at which $u^r=0$) in a proper time $\Delta\tau=r_g/c$.
The radial motion of a massive particle (in terms of the proper time) in the spacetime (\ref{interv1}) for $r\le r_g/4$ is the image (in the sense of the method of image charges for spheres in electrostatics) of the particle's motion for $r\ge r_g/4$.
Applying the transformation (\ref{transf}) to Eqs. (\ref{mot1}) and (\ref{mot2}) in the region $r\le r_g/4$ gives
\begin{eqnarray}
& & \frac{dt}{d\tau}=\sqrt{h_0}/h, \label{motin1} \\
& & \frac{dr'}{cd\tau}=-\epsilon(h_0 h^{-1} f^{-1}-f^{-1})^{1/2},
\label{motin2}
\end{eqnarray}
where $h=(1-r_g/(4r'))^2/(1+r_g/(4r'))^2$ and $f=(1+r_g/(4r'))^4$.
An infalling radial geodesic motion inside a black hole appears, in terms of the new radial coordinate $r'$, as an outgoing motion from a white hole (the time reversal of a black hole).

The local velocity of the particle $v_l$, measured in terms of the proper time, as determined by static clocks synchronized along the trajectory of the particle, is related to $u^0$ by $u^0=(h(1-v_l^2/c^2))^{-1/2}$ \cite{LL2}.
As the particle moves from $r=r_0$ to $r=r_g/4$, $v_l$ increases from zero to $c$, and as the particle moves from $r=r_g/4$ to $r=r_1$, $v_l$ decreases to zero.
In a Schwarzschild field, $v_l$ exceeds $c$ inside a black hole, which does not violate Einstein's theory of relativity because the interior of a Schwarzschild black hole is not static and neither can be clocks synchronized along the trajectory of the particle.

Consider radially moving geodesic clocks that are falling into a black hole from the isotropic radius $r_0$, and synchronized such that the time of each clock at the instant of release equals the proper time $\tau_0$ of a clock at rest that remains fixed at $r_0$.
The time at any event is taken to be equal to the proper time on the radially falling clock that is coincident with this event, following the procedure due to Gautreau and Hoffmann \cite{GH}.
Equations (\ref{mot1}) and (\ref{mot2}) give (for $\epsilon=-1$)
\begin{equation}
cd\tau=\sqrt{h_0}cdt+\sqrt{f(h_0-h)/h}dr,
\label{mot5}
\end{equation}
which, using $\tau_0=\sqrt{h_0}t_0$, integrates to
\begin{equation}
c\tau=\sqrt{h_0}ct+\int_{r_0}^{r}\sqrt{f(h_0-h)/h}dr,
\label{mot6}
\end{equation}
giving the transformation between the coordinates $(r,t)$ and $(r,\tau)$.
In terms of $\tau$, the metric (\ref{interv1}) becomes
\begin{equation}
ds^2=h(cd\tau-\sqrt{f(h_0-h)/h}dr)^2/h_0-fdr^2-fr^2d\Omega^2.
\label{interv2}
\end{equation}
Radial null geodesics are given by $ds=0$ and $d\Omega=0$:
\begin{equation}
\frac{cd\tau}{dr}=\sqrt{fh_0/h}(\sqrt{1-h/h_0}\pm1).
\label{nul}
\end{equation}
The plus (minus) sign corresponds to an outgoing (infalling) null geodesic.
For $r\rightarrow\infty$, the spacetime is Galilean and $cd\tau/dr=\pm1$.
For $r=r_g/4$, $cd\tau/dr=\infty$ for the outgoing null geodesic and $0$ for the infalling one, as shown in Fig.~\ref{fig1}.
For $r=0$, $cd\tau/dr=\pm\infty$; however, in terms of the new radial coordinate $r'$ (\ref{transf}), we obtain $cd\tau/dr'=\pm1$ (Galilean spacetime).
Massive particles can move in both radial directions, except at the unidirectional surface $r=r_g/4$, where only infalling geodesics (decreasing $r$) lie inside the light cone.
\begin{figure}[ht]
\centering
\includegraphics[width=4in]{lightcones.eps}
\caption{Light cones in the gravitational field represented by the metric (\ref{interv2}).}
\label{fig1}
\end{figure}

The singularity theorem of Penrose guarantees that, whenever matter satisfies reasonable energy conditions, some sort of geodesic incompleteness occurs inside a black hole \cite{Wald}.
The key component in this theorem is the Raychaudhuri equation \cite{Ray}, describing the time evolution of the expansion scalar for a timelike congruence.
The expansion scalar $\theta=u^i_{\phantom{i};i}$ measures the fractional rate at which a small volume of matter changes with respect to time as measured by a comoving observer.
The Raychaudhuri equation guarantees that any timelike geodesic inside a Schwarzschild black hole converges in a caustic ($\theta\rightarrow -\infty$) within a finite proper time.
For the infalling radial motion into the Einstein-Rosen bridge (\ref{interv1}) from infinity ($h_0=1$), Eq. (\ref{mot2}) yields
\begin{equation}
\theta=\frac{1}{\sqrt{hf^3}r^2}(\sqrt{hf^3}r^2 u^r)_{,r}=-\frac{3}{2}\sqrt{\frac{r_g}{r^3}}\frac{\textrm{sgn}(1-r_g/(4r))}{(1+r_g/(4r))^3}.
\label{theta}
\end{equation}
As the particle moves from infinity to $r=r_g/4$, $\theta$ decreases from $0$ to $-3/(2r_g)$.
At $r=r_g/4$, $\theta$ undergoes a discontinuity, jumping to $3/(2r_g)$.
As the particle moves from $r=r_g/4$ to $r=0$, $\theta$ decreases back to zero.
The discontinuity of the expansion scalar at the event horizon of a wormhole prevents $\theta$ from decreasing to $-\infty$, as it occurs inside a Schwarzschild black hole.
Therefore this discontinuity guarantees that timelike geodesics in the gravitational field of a wormhole are complete.

\section{Kruskal representation}

A maximal extension of the Schwarzschild metric \cite{Syn,KS} shows that a massive particle cannot travel in the spacetime of a Schwarzschild black hole from Kruskal's right-hand quadrant (exterior region I in Fig.~\ref{fig2}) to left-hand quadrant (exterior region III) without violating causality: the Schwarzschild bridge is not traversable \cite{FW,BMT}.
Such a particle either remains in region I or moves to the upper quadrant (interior region II), where it reaches the central singularity and its proper time ends.
Equation (\ref{mot3}) can be solved in the Schwarzschild coordinates, where $h=1-r_g/r_S$ and $f=(1-r_g/r_S)^{-1}$, in the parametric form \cite{Chan}:
\begin{eqnarray}
& & \frac{r_S}{r_{S0}}=\frac{1}{2}(1+\textrm{cos}\eta), \label{solut1} \\
& & \frac{ct}{r_g}=\textrm{ln}\frac{k+\textrm{tan}(\eta/2)}{k-\textrm{tan}(\eta/2)}+k\Bigl(\eta+\frac{r_{S0}}{2r_g}(\eta+\textrm{sin}\eta)\Bigr),
\label{solut2}
\end{eqnarray}
where $r_{S0}=r_0(1+r_g/(4r_0))^2$, $k=(r_{S0}/r_g-1)^{1/2}$, and the parameter $\eta$ goes from $0$ (the particle is at rest at $t=0$) to the value at the event horizon, $\eta_h=2\textrm{tan}^{-1}k$.
Equations (\ref{mot2}) and (\ref{solut1}) give the proper time $\tau$ as a monotonously increasing function of $\eta$:
\begin{equation}
\tau=\frac{1}{2c}\sqrt{\frac{r_{S0}^3}{r_g}}(\eta+\textrm{sin}\eta).
\label{prop}
\end{equation}
Transforming from the Schwarzschild coordinates $(r,t)$ to the Kruskal coordinates $(U,V)$ (in region I) \cite{KS},
\begin{eqnarray}
& & U=(r_S/r_g-1)^{1/2}e^{r_S/(2r_g)}\textrm{cosh}\frac{ct}{2r_g}, \label{Kr1} \\
& & V=(r_S/r_g-1)^{1/2}e^{r_S/(2r_g)}\textrm{sinh}\frac{ct}{2r_g},
\label{Kr2}
\end{eqnarray}
together with Eq. (\ref{mot3}), gives the derivative $dV/dU$:
\begin{equation}
\frac{dV}{dU}=\frac{k-\textrm{tanh}\frac{ct}{2r_g}\cdot(\frac{r_{S0}}{r_S}-1)^{1/2}}{k\textrm{tanh}\frac{ct}{2r_g}-(\frac{r_{S0}}{r_S}-1)^{1/2}}.
\label{der1}
\end{equation}
In the limit $\eta\rightarrow\eta_h$ we can write $\eta=\eta_h-\xi$, where $0<\xi\ll 1$, which gives $r_S/r_g-1\approx k\xi$, $\textrm{exp}(ct/(2r_g))\approx 2\textrm{cos}(\eta_h/2)\sqrt{k/\xi}\textrm{exp}[k(\eta_h+r_{S0}(\eta_h+\textrm{sin}\eta_h)/(2r_g))/2]$ and $(r_{S0}/r_S-1)^{1/2}\approx k-\xi/2$.
We obtain the values of $U,V$ for the particle at the event horizon:
\begin{equation}
U_h=V_h=\frac{\sqrt{e}k}{\sqrt{1+k^2}}e^{\frac{k}{2}\big((k^2+3)\textrm{tan}^{-1}k+k\big)},
\label{KrEH}
\end{equation}
and the value of the slope $dV/dU$ at the horizon:
\begin{equation}
\frac{dV}{dU}\Big|_h=\zeta=\textrm{coth}\Big[\frac{k}{2}\Big((k^2+3)\textrm{tan}^{-1}k+k\Big)\Big].
\label{der2}
\end{equation}

After the particle, which started moving from the radius $r_{S0}>r_g$ at $t=0$ (point $A$ in Fig.~\ref{fig2}), reaches the event horizon of a Schwarzschild black hole at $r_S=r_g,\,r=r_g/4$ (point $B$), it moves toward the central singularity at $r_S=0$ (point $E$) as $\eta$ (which is an increasing function of the particle's proper time $\tau$) goes to $\pi$.
\begin{figure}[ht]
\centering
\includegraphics[width=4in]{trajectory.eps}
\caption{Infalling radial geodesic motion of a massive particle into a Schwarzschild black hole ($ABE$) and an Einstein-Rosen black hole ($ABCD$) in the Kruskal coordinates.
For the Einstein-Rosen black hole, the events on the segment $OF$ are identified with the events on the segment $OG$, e.g., points $B$ and $C$ represent the same spacetime event.}
\label{fig2}
\end{figure}

The completeness of the particle's geodesic in the spacetime of the Einstein-Rosen bridge in the Kruskal coordinates can be explained if we note that both coordinate pairs $(-U,-V)$ and $(U,V)$ mathematically correspond to the same coordinates $r,t$.
Rindler suggested that both coordinate pairs $(-U,-V)$ and $(U,V)$ physically represent the same coordinates $r,t$, i.e. region III is identical with region I, and (interior) region IV is identical with region II; the Kruskal spacetime is elliptic \cite{Rin}.
In order to represent the Einstein-Rosen bridge in the Kruskal coordinates, we need to impose Rindler's elliptic identification of event antipodes {\em only} on the line $V=U$, i.e. only the two antipodal future event horizons are identical.
After the particle reaches the event horizon of an Einstein-Rosen black hole at point $B$, it moves from point $C$, which is {\em identical} with point $B$, to point $D$, which is related to point $A$ via the transformation (\ref{transf}) of the isotropic radial coordinate $r$.
As the particle moves from point $C$ to $D$, the proper time $\tau$ increases, while the coordinate time $t$ decreases (runs in the reverse direction with respect to observers in region I).
This reversion does not cause any problems, because it occurs after $t\rightarrow\infty$, so observers in region I never see it.
At infinity of region III, the proper time $\tau$ of the particle coincides with $-t$, so $-t$ has the meaning of the physical coordinate time of a white hole (region III).

The dependence of $r$ on $\eta$ in the radial geodesic motion of a massive particle into an Einstein-Rosen black hole is given by
\begin{equation}
r\Bigl(1+\frac{r_g}{4r}\Bigr)^2=\frac{1}{2}(1+\textrm{cos}\eta)r_0 \Bigl(1+\frac{r_g}{4r_0}\Bigr)^2.
\label{solut3}
\end{equation}
The segment $AB$ corresponds to the motion from $r=r_0$ ($\tau=0$, $dV/dU=\infty$) to $r=r_g/4$ ($\tau=\tau_h=\sqrt{r_{S0}^3/r_g}(\eta_h+\textrm{sin}\eta_h)/(2c)$, $dV/dU=\zeta$), and the segment $CD$ corresponds to the motion from $r=r_g/4$ ($\tau=\tau_h$, $dV/dU=\zeta$; the slope $dV/dU$ is continuous at the horizon as we go from point $B$ to $C$) to $r=r_1=r_g^2/(16r_0)$ ($\tau=2\tau_h$, $dV/dU=\infty$).
Therefore crossing the Einstein-Rosen bridge is possible if we regard Rindler's elliptic identification of the two antipodal future event horizons as physical.
The particle enters region III (with no possibility of coming back to region I), which has no curvature singularities and is mathematically equivalent to the asymptotically flat exterior of a white hole \cite{NN}.
The particle's proper time does not end in this region, but extends to infinity.

\section{Discussion}

The difference in the character of the radial motion inside the event horizon between an Einstein-Rosen black hole and a Schwarzschild black hole indicates that the two black hole solutions are physically different with regard to the nature of the interior sheet and equivalent with regard to the nature of the static exterior sheet (region I).
For an Einstein-Rosen black hole with Rindler's elliptic identification of the two antipodal future event horizons the interior (from the point of view of a distant observer; it really is the exterior region III) is static, while for a Schwarzschild black hole the interior (region II) is nonstatic.
Rindler's identification also guarantees that the Einstein-Rosen bridge is a stable solution to the gravitational field equations; without this identification the bridge would be unstable \cite{FW}.
Another difference between a Schwarzschild black hole and an Einstein-Rosen black hole comes from the integral of the time-time component of the gravitational energy-momentum pseudotensor (either Einstein or Landau-Lifshitz) over the interior hypersurface, i.e. the total energy of the system \cite{LL2,Schr}.
This energy equals the sensible physical value $r_gc^4/2G=Mc^2$ for an Einstein-Rosen black hole, but diverges for a Schwarzschild black hole.
Although the construction of the Einstein-Rosen bridge in the Kruskal coordinates through Rindler's elliptic identification of the two antipodal future event horizons may seem artificial, the necessity for Rindler's identification to avoid the particle's motion to the central singularity may be related to a singular behavior of the Kruskal metric in the Minkowski limit $r_g\rightarrow0$.
However, the Kruskal coordinates are advantageous in describing the Einstein-Rosen bridge because the surface $r=r_g/4$ is regular ($\textrm{det}\,g_{\mu\nu}<0$) in these coordinates.

The Schwarzschild black hole solution, singular at the center, does not exist in isotropic coordinates, while the Einstein-Rosen bridge (wormhole) geometry, regular everywhere, can be built in the Schwarzschild coordinates by gluing together two Schwarzschild exterior sheets at their common event horizon \cite{V}.
While the Schwarzschild metric is the spherically symmetric solution to the Einstein field equations in vacuum if we solve these equations using the Schwarzschild coordinates, the Einstein-Rosen bridge is the spherically symmetric solution to the Einstein field equations if we solve these equations using isotropic coordinates for a source which is vacuum everywhere except at the surface $r=r_g/4$.
It has been shown that the Einstein-Rosen bridge metric (\ref{interv1}) is not a solution of the vacuum Einstein equations but it requires the presence of a nonzero energy-momentum tensor source $T_{\mu\nu}$ that is divergent and violates the energy conditions at the throat of the wormhole \cite{V,Gue}.
This metric satisfies the Levi-Civita identity, $R^0_0=\frac{1}{\sqrt{g_{00}}}\nabla^2(\sqrt{g_{00}})$, where $\nabla^2$ is the Laplace-Beltrami operator \cite{Gue,Fra}.
This identity gives for (\ref{interv1}):
\begin{equation}
R^0_0=\frac{8\pi G}{c^4}\biggl(T^0_0-\frac{1}{2}T\biggr)=\frac{\delta(r-r_g/4)}{8|r-r_g/4|}=\frac{\delta\Bigl((r-r_g/4)^2\Bigr)}{4}.
\end{equation}
The curvature scalar at the throat of such a wormhole acquires a similar delta-function contribution.
It has been shown in \cite{Gue} that the delta-function matter source in the Einstein-Rosen bridge at $r=r_g/4$ is a lightlike brane self-consistently interacting with gravity.

Both black hole solutions are mathematically legitimate, and only experiment or observation can reveal the nature of the infalling radial motion of a particle into a physical black hole.
Since the two solutions are indistinguishable for distant observers, which can see only the exterior sheet, the nature of the interior of a physical black hole cannot be satisfactorily determined, unless an observer enters or resides in the interior region.
This condition would be satisfied if our Universe were the interior of a black hole existing in a bigger universe \cite{Pat}.
Because Einstein's general theory of relativity does not choose a time orientation, if a black hole can form from the gravitational collapse of matter through an event horizon in the future then the reverse process also possible.
Such a process would describe an exploding white hole: matter emerging from an event horizon in the past, like the expanding Universe.
Scenarios in which the Universe was born from the interior of an Einstein-Rosen black hole may avoid many of the problems of the standard Big-Bang cosmology and the black hole information-loss problem \cite{EB}.
These scenarios, involving gravitational collapse of a sphere of dust in isotropic coordinates, and generalization of the results of the present Letter to Schwarzschild-de Sitter and Kerr black holes will be the subjects of further study.

redpajama_set_name: RedPajamaArXiv