text:
\section{ Introduction}
Key-based enciphering is a well-adopted technique to ensure the
security in current data transmission system. However, for secure
communications in wireless networks, the distributions and
managements of secret keys may be challenging tasks
\cite{Liang_same_marginal}. The physical-layer security introduced
in \cite{csiszar1978broadcast}\cite{Wyner_wiretap} is appealing
due to its keyless nature. The basic building block of
physical-layer security is the so-called wiretap channel. In this
channel, a source node wants to transmit confidential messages
securely to a legitimate receiver and to keep the eavesdropper as
ignorant of the message as possible. Wyner \cite{Wyner_wiretap}
characterized the secrecy capacity of the discrete memoryless
wiretap channel, in which the secret key was not used. The secrecy
capacity is the largest secrecy rate of communication between the
source and the destination nodes with the eavesdropper knowing no
information of the messages. In order to meet the demands of high
data rate transmissions and improve the connectivities of the
secure networks \cite{Secureconnect}, the multiple antenna systems
with security concerns are considered by several authors. In
\cite{Shafiee_secrecy_2_2_1_J}, the secrecy capacity of a Gaussian
channel with two-input, two-output, single-antenna-eavesdropper
was first characterized. This result was extended by
\cite{Khisti_MIMOME}, in which the secrecy capacities of general
Gaussian multiple-input multiple-output
multiple-antenna-eavesdropper (MIMOME) channels are proved. In
wireless environments, with perfect knowledge of the legitimate
receiver's channel state information at the transmitter (CSIT),
the time-varying characteristic of fading eavesdropper channels
can be further exploited to enhance the secrecy
\cite{gopala2008secrecy,Li_fading_secrecy_j,CollingsSecrecy,Secrecy_scJSAC}.

However, to attain the secrecy capacity results in
\cite{Shafiee_secrecy_2_2_1_J,Khisti_MIMOME,gopala2008secrecy}, at
least the perfect knowledge of the legitimate receiver's CSIT is
required. For the fast fading channels, it may be hard to track
the rapidly varying channel coefficients because of the limited
feedback bandwidth and the delay caused by the channel estimation.
Thus for fast-fading channels, it is more practical to consider
the case with only partial CSIT of the legitimate channel. For the
setting where only statistical CSIT of both legitimate and
eavesdropper channels is available, the secrecy capacity is only
rigorously characterized for multiple-input single-output
single-antenna-eavesdropper (MISOSE) Rayleigh fast-faded channels
\cite{Secrecy_sc}. A negative phenomenon, revealed in
\cite{Secrecy_sc}, showed that for the MISOSE channels with only
statistical CSIT, the secrecy capacities would neither scale with
the number of transmit antennas nor the signal-to-noise ratio
(SNR). Thus using multiple transmitter antennas in the MISOSE
system limitedly helps increase the secrecy capacity, compared
with the system using single transmitter antenna.

In this paper, we want to overcome the aforementioned drawbacks of
the MISOSE system. We then focus on the MIMOME fast Rayleigh-faded
channels where the transmitter only have the statistical CSIT of
the legitimate and eavesdropper channels. Two different power
constraints are considered. One is the total power constraint over
all transmit antennas, and the other is the more practical
per-antenna power constraint. Under the total power constraint
over all transmit antennas, the MIMOME secrecy capacity is
characterized when antennas of the legitimate receiver are more
than (or equal to) those of the eavesdropper. To the best of the
author's knowledge, this is the first MIMOME secrecy capacity
result with partial CSIT. Compared with our previous works
\cite{Secrecy_sc}, new proof techniques are developed for the
MIMOME channels. First, we establish a new secrecy capacity
upper-bound by dividing the channel matrix of legitimate receiver
into two submatrices, one of the submatrices has dimensions equal
to those of the eavesdropper's channel matrix, to form an MIMOME
degraded channel. Second, instead of using completely monotone
property as \cite{Secrecy_sc}, which may only exists for the
MISOSE problem, we solve the stochastic MIMOME optimization
problem by transforming it to an equivalent Schur-concave problem.
The key to this transformation is using the proposed equivalent
MIMOME degraded channel. According to our secrecy capacity
results, on the contrary to \cite{Secrecy_sc}, we observe that the
SNR scaling of secrecy capacity can be obtained in the MIMOME
channels when the number of antenna of the legitimate receiver is
larger than (but not equal to) that of the eavesdropper. Also when
the number of transmit antennas is fixed, increasing the
difference between numbers of antennas of legitimate receiver and
eavesdropper helps to increase the secrecy capacities.

For the cases where the transmitters are subject to the practical
per-antenna power constraints, we also fully characterize the
secrecy capacities for the MISOSE wiretap channels with
statistical CSIT. Because the optimization problem subject to the
per-antenna power constraint is not Schur-concave, only numerical
algorithm is developed to find the secrecy capacity for the MIMOME
channel. However, we can still show that under this constraint,
when there are sufficient transmit antennas with large enough
transmitted power each, the MIMOME secrecy capacities can also
scale with SNRs. Such a scaling cannot be obtained for the MISOSE
wiretap channels under per-antenna power constraint, even when all
transmit antennas are allowed to transmit with large power.

Under total power constraint over all transmit antennas, several
works have studied the secrecy capacities in various channel
settings. For channels with full CSIT, the secrecy capacities were
found in \cite{Shafiee_secrecy_2_2_1_J,Khisti_MIMOME}; and for
ergodic slow fading channels with partial CSIT, they were found in
\cite{gopala2008secrecy}. However, for fast fading channels with
full legitimate CSIT and statistical eavesdropper CSIT, the
secrecy capacities are unknown. And several works instead studied
the achievable secrecy rate in these channels
\cite{Li_fading_secrecy_j,CollingsSecrecy,Secrecy_scJSAC}. The
settings in
\cite{Shafiee_secrecy_2_2_1_J,Khisti_MIMOME,gopala2008secrecy,Li_fading_secrecy_j,CollingsSecrecy,Secrecy_scJSAC}
are fundamentally different to ours. In this paper, fast fading
channels with only statistical CSIT of both legitimate receiver
and eavesdropper are considered. Our works are the first secrecy
capacity results for MIMOME channels with statistical CSIT. More
related works and the comparisons with our works can be found in
our MISOSE previous work \cite{Secrecy_sc} and references within.
Subject to the per-antenna power constraint, secrecy rate
optimization with full CSIT was studied in \cite{li2010optimal}.
Besides the difference between CSIT assumptions compared to ours,
the channel input matrix in \cite{li2010optimal} was optimized
numerically and the optimality was not guaranteed. However, in our
work, optimal channel input covariance matrix for the MISOSE
channel is analytically solved, and our numerical algorithm for
the MIMOME channel can guarantee the optimality.

The rest of the paper is organized as follows. In Section
\ref{Sec_system_model} we introduce the considered system model.
In Section \ref{Sec_total_power}, under total power constraint
over all tranmitter antennas, we prove the secrecy capacities for
fast fading MIMOME wiretap channels with statistical CSIT. In
Section \ref{Sec_per_antenna}, we investigate secrecy capacities
under per-antenna power constraints. Simulation results are
provided in Section \ref{Sec_simulation}, and the conclusion is
given in Section \ref{Sec_conclusion}.

\underline{\bf{Notations}}: In this paper, lower and upper case
bold alphabets denote vectors and matrices, respectively. The
zero-mean complex Gaussian random vector with covariance matrix
$\Sigma$ is denoted as $CN(0, \Sigma)$. The mutual information
between two random vectors ${\mathbf x}$ and ${\mathbf y}$ is $I({\mathbf x};{\mathbf y})$, while
the conditional differential entropy is $h({\mathbf x}|{\mathbf y})$. The
superscript $(.)^H$ denotes the transpose complex conjugate.
$|\mathbf{A}|$ and $|a|$ represent the determinant of the square
matrix $\mathbf{A}$ and the absolute value of the scalar variable
$a$, respectively. The trace of $\mathbf{A}$ is denoted by
${\rm tr}(\mathbf{A})$. The element of ${\mathbf A}$ in the $i$th row and $j$th
column is $\{{\mathbf A}\}_{i,j}$. A diagonal matrix whose diagonal
entries are $a_1 \ldots a_k$ is denoted by $diag(a_1 \ldots a_k)$.
The positive semidefinite ordering between Hermitian matrices
${\mathbf A}$ and ${\mathbf B}$ are denoted by ${\mathbf A}\succeq {\mathbf B}$ (${\mathbf A}\succ {\mathbf B}$),
where ${\mathbf A}-{\mathbf B}$ is a positive semi-definite (definite) matrix.

\section{System Model}\label{Sec_system_model}
In the considered MIMOME wiretap channel, we study the problem of
reliably communicating a secret message $w$ from the transmitter
to the legitimate receiver, subject to a constraint on the
information attainable by the eavesdropper (in upcoming
\eqref{eq_equivocation_given_h}). The transmitter has $n_t$
antennas, while the legitimate receiver and eavesdropper
respectively have $n_r$ and $n_e$ antennas as
\begin{align}
{\mathbf y}&={\mathbf H}\mathbf{x}+{\mathbf n}_{y}, \label{EQ_main_MISOSE} \\
{\mathbf z}&={\mathbf G}\mathbf{x}+{\mathbf n}_{z},\label{EQ_Eve_MISOSE}
\end{align}
where $\mathbf{x} \in \mathds{C}^{n_t \times 1}$ represents the
transmitted vector signal; the legitimate channel matrix is ${\mathbf H}
\in \mathds{C}^{n_r \times n_t}$ while the eavesdropper channel
matrix is ${\mathbf G} \in \mathds{C}^{n_e \times n_t}$; ${\mathbf n}_y$ and
${\mathbf n}_z$ are additive white Gaussian noise vectors at the
legitimate receiver and eavesdropper, respectively, with each
element independent and identically distributed (i.i.d.),
circularly symmetric, and having zero mean and unit variance. The
channels are assumed to be fast Rayleigh fading, that is, each
element of ${\mathbf H}$ and ${\mathbf G}$ is i.i.d distributed as
\begin{equation} \label{eq_Rayleigh}
CN(0, \sigma_{{\mathbf h}}^2) \;\; \mbox{and} \;\; CN(0, \sigma_{{\mathbf g}}^2),
\end{equation}
respectively, while the channel coefficients change in each symbol
time. The $\mathbf{H}$, $\mathbf{G}$, ${\mathbf n}_y$ and ${\mathbf n}_z$ are
independent. We assume that the legitimate receiver knows the
instantaneous channel state information of $\mathbf{H}$ perfectly,
while the eavesdropper knows those of $\mathbf{H}$ and
$\mathbf{G}$ perfectly. As for the CSIT, only the distributions of
$\mathbf{H}$ and $\mathbf{G}$ are known while the realizations of
$\mathbf{H}$ and $\mathbf{G}$ are unknown at the transmitter.

In this work, we consider two kinds of constraints for the channel
input ${\mathbf x}$. The first one is the total power constraint over all
transmitter antennas  as
\begin{equation} \label{power_cons}
\mathrm{Tr}(\Sigma_{{\mathbf x}}) \leq P,
\end{equation}
where $\Sigma_{{\mathbf x}}$ is the covariance matrix of $\mathbf{x}$.
Note that under \eqref{power_cons}, the transmitter can perform
power allocation between transmitter antennas to increase the
secrecy capacity. In addition to \eqref{power_cons}, we also
consider a more practical per-antenna constraint for the
transmitter antennas as
\begin{equation} \label{eq_idv_power_cons}
\{\Sigma_{\mathbf x}\}_{ii}\leq P_i,
\end{equation}
for $i=1,\ldots, n_t$. Note that the per-antenna constraint
\eqref{eq_idv_power_cons} is more stringent than the total power
constraint \eqref{power_cons} when $\Sigma_{i=1}^{N_t} P_i \leq
P$.

The perfect secrecy and the corresponding secrecy capacity are
defined as follows. Consider a $(2^{NR}, N)$-code with an encoder
that maps the message $w\in {\mathcal W}_N=\{1,2,\ldots, 2^{NR}\}$ into a
length-$N$ codeword, and a decoder at the legitimate receiver that
maps the received sequence $y^N$ (the collections of $y$ over the
code length $N$) from the legitimate channel
\eqref{EQ_main_MISOSE} to an estimated message $\hat w\in{\mathcal W}_N$.
We then have the following definitions, where ${\mathbf z}^N$,
$\mathbf{H}^N$, and $\mathbf{G}^N$ are the collections of ${\mathbf z}$,
$\mathbf{H}$, and $\mathbf{G}$ over the code length $N$,
respectively.
\begin{Def}[Secrecy Capacity
\cite{Liang_same_marginal}\cite{Wyner_wiretap}\cite{gopala2008secrecy}]
\label{Def_Perfect} {\it Perfect secrecy is achievable with rate
$R$ if, for any $\varepsilon>0$, there exists a sequence of
$(2^{NR}, N)$-codes and an integer $N_0$ such that for any $N>N_0$
\begin{align}
&{R_e=h(w|{\mathbf z}^N,\mathbf{H}^N, \mathbf{G}^N)/N \geq R-\varepsilon},
\label{eq_equivocation_given_h} \\ \mathrm{and} \;\;\; &{\rm
Pr}(\hat{w}\neq w) \leq \varepsilon, \notag
\end{align}
where { $R_e$ in \eqref{eq_equivocation_given_h} is the
equivocation} rate and $w$ is the secret message. The {\bf secrecy
capacity} is the supremum of all achievable secrecy rates.}
\end{Def}
Note that the perfect secrecy requirement in
\eqref{eq_equivocation_given_h} is measured by $I(w;
{\mathbf z}^N,\mathbf{H}^N, \mathbf{G}^N)/N=R-R_e$, which is based on all
the observations $({\mathbf z}^N,\mathbf{H}^N, \mathbf{G}^N)$ the
eavesdropper has.

From Csisz{\'{a}}r and K{\"{o}}rner's seminal work
\cite{csiszar1978broadcast}, we know that the secrecy capacity of
MIMOME channel \eqref{EQ_main_MISOSE} and \eqref{EQ_Eve_MISOSE} is
\begin{align}
&\underset{U}{\max}\;\;I(U;{\mathbf y},{\mathbf H})-I(U;{\mathbf z},{\mathbf H},{\mathbf G}), \notag \\
=&\underset{U}{\max}\;\;I(U;{\mathbf y}|{\mathbf H})-I(U;{\mathbf z}|{\mathbf H},{\mathbf G}),
\label{EQ_partial_CSI_rate}
\end{align}
where $U$ is an auxiliary random variable satisfying the Markov
relationship $U\rightarrow {\mathbf x} \rightarrow ({\mathbf y},{\mathbf H}), ({\mathbf z},
{\mathbf H},{\mathbf G})$, and \eqref{EQ_partial_CSI_rate} results from the fact
that the transmitter does not have the knowledge of the
realizations of ${\mathbf H}$ and ${\mathbf G}$. However, the optimal choice of
$U$ which maximizes the secrecy capacity of considered fast fading
MIMOME channel is \textit{unknown}. In this paper, we want to
fully characterize the optimal $U$. Note that the secrecy capacity
considered here is achieved by encoding over multiple channel
states and the perfect secrecy constraint must be satisfied for
all $N>N_0$. This implies that no secrecy outage
\cite{yuksel2011diversity} \cite{CollingsSecrecy} is allowed. In
delay-limited applications, such perfect secrecy condition may not
be achievable and, thus, a tradeoff exists between secrecy rate
and secrecy outage probability. These issues have been discussed
in \cite{yuksel2011diversity} \cite{CollingsSecrecy} and are
beyond the scope of this paper.

\section{Secrecy capacity under the \\ total power constraint}\label{Sec_total_power}
In this section, we explicitly find the optimal $U$ in
\eqref{EQ_partial_CSI_rate} for wiretap channel
\eqref{EQ_main_MISOSE}\eqref{EQ_Eve_MISOSE}, and fully
characterize the MIMOME secrecy capacity with statistical CSIT in
the upcoming Theorem \ref{Theorem_No_CQI}. When there is full CSIT
\cite{Khisti_MIMOME}, one can find the optimal auxiliary random
variable $U$ by constructing an equivalent degraded MIMOME channel
to upper-bound the secrecy capacity. However, with only
statistical CSIT, if one naively applies the degraded channel
construction method in \cite{Khisti_MIMOME}, the resulting secrecy
capacity upper bound will depend on the realizations of
(${\mathbf H}$,${\mathbf G}$) and become very loose. Thus in general it is very
hard to find the optimal $U$ maximizing
\eqref{EQ_partial_CSI_rate}

Due to the difficulty mentioned in the previous paragraph, with
statistical CSIT, finding the optimal $U$ maximizing
\eqref{EQ_partial_CSI_rate} is very hard in general. However, in
the following Lemma, for the special cases where the MIMOME
channels are Rayleigh faded as \eqref{eq_Rayleigh}, we show that
one may still construct a degraded channel for upper-bounding and
find the optimal $U$ maximizing \eqref{EQ_partial_CSI_rate}. The
key for building this equivalent degraded MIMOME channel for
\eqref{EQ_main_MISOSE}\eqref{EQ_Eve_MISOSE} is replacing the
legitimate channel ${\mathbf H}$ with equivalent ${\mathbf H}'$ in upcoming
\eqref{eq_H_divide} as follows. When $n_r \geq n_e$, one can
divide the legitimate channel matrix ${\mathbf H}$ in
\eqref{EQ_main_MISOSE} as two submatrices
\begin{equation} \label{eq_split_H}
{\mathbf H}=[{\mathbf H}^T_{(n_r-n_e)} \; {\mathbf H}^T_{n_e}]^T,
\end{equation}
where ${\mathbf H}_{(n_r-n_e)} \in \mathds{C}^{(n_r-n_e) \times n_t}$ and
${\mathbf H}_{n_e} \in \mathds{C}^{n_e \times n_t}$, with each element of
${\mathbf H}_{(n_r-n_e)}$ and ${\mathbf H}_{n_e}$ distributed as i.i.d. Gaussian
$CN(0,\sigma_h^2)$. From the properties of complex Gaussian
distributions, the distribution of ${\mathbf H}$ is the same as that of
\begin{equation} \label{eq_H_divide}
{\mathbf H}'=[{\mathbf H}^T_{(n_r-n_e)} \;
\left(\frac{\sigma_{{\mathbf h}}}{\sigma_{{\mathbf g}}}\right){\mathbf G}^T]^T,
\end{equation}
because each element of eavesdropper channel matrix ${\mathbf G}$ is
distributed as $CN(0,\sigma_g^2)$ according to
\eqref{eq_Rayleigh}. With \eqref{eq_H_divide}, one can build an
equivalent degraded MIMOME channel for
\eqref{EQ_main_MISOSE}\eqref{EQ_Eve_MISOSE} (see upcoming
\eqref{eq_same_dist_leg_channel} \eqref{eq_degraded_markov}),
where the received signal at eavesdropper can be treated as a
degraded version of that at the legitimate receiver. Then we can
obtain a tight secrecy capacity upper bound and have
the following property of the optimal $U$ for \eqref{EQ_partial_CSI_rate}. \\
\begin{lemma} \label{Lem_No_CQI}
For the MIMOME fast Rayleigh fading wiretap channel
\eqref{EQ_main_MISOSE}\eqref{EQ_Eve_MISOSE} with the statistical
CSIT of $\mathbf{H}$ and $\mathbf{G}$, using Gaussian ${\mathbf x}$
without prefixing $U \equiv {\mathbf x}$ is the optimal transmission
strategy for \eqref{EQ_partial_CSI_rate} under \eqref{power_cons}
when $n_r \geq n_e$ and $\sigma_h \geq \sigma_g$, where $n_r$ and
$n_e$ respectively are the number of antennas at the legitimate
receiver and eavesdropper, while $\sigma_h$ and $\sigma_g$
respectively are the variances of the legitimate and eavesdropper
channels as
\eqref{eq_Rayleigh}.\\
\end{lemma}

\begin{proof}
We first form the degraded MIMOME channel with respect to
\eqref{EQ_main_MISOSE}\eqref{EQ_Eve_MISOSE}. Here we will only
consider the cases where $\sigma^2_g>0$, since the case with
$\sigma^2_g=0$ is a trivial one which corresponds to channel
without eavesdropper. Note that ${\mathbf H}$ in \eqref{eq_split_H} has
the same distribution as ${\mathbf H}'$ in \eqref{eq_H_divide}. Then we
can form an equivalent received signal
\begin{equation} \label{eq_same_dist_leg_channel}
{\mathbf y}'={\mathbf H}'{\mathbf x}+{\mathbf n}_y,
\end{equation}
which has the same marginal distribution as the legitimate
received signal ${\mathbf y}$ in \eqref{EQ_main_MISOSE}. Now, rewrite the
above ${\mathbf y}'$ using \eqref{eq_H_divide} as
\begin{align}
{\mathbf y}'&=[{\mathbf H}_{(n_r-n_e)}^T \;
\left(\frac{\sigma_{{\mathbf h}}}{\sigma_{{\mathbf g}}}\right){\mathbf G}^T]^T\mathbf{x}+{\mathbf n}_{y}\notag
\\ &=\left[\left({\mathbf y}'_{(n_r-n_e)}\right)^T\;\left({\mathbf y}'_{n_e}\right)^T\right]^T, \label{eq_same_dist_channel_nor}
\end{align}
where ${\mathbf y}'_{(n_r-n_e)} \in \mathds{C}^{(n_r-n_e) \times 1}$ and
${\mathbf y}'_{n_e} \in \mathds{C}^{n_e \times 1}$. Now we consider
\[
\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}'=\left[\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}\left({\mathbf y}'_{(n_r-n_e)}\right)^T\;\left({\mathbf G}{\mathbf x}+(\sigma_{{\mathbf g}}/\sigma_{{\mathbf h}}){\mathbf n}_{y,n_e}\right)^T\right]^T,
\notag
\]
where the noise vector ${\mathbf n}_{y,n_e} \in \mathds{C}^{n_e \times 1}$
comes from dividing the noise vector at the legitimate receiver as
${\mathbf n}_y=[{\mathbf n}^T_{y,(n_r-n_e)} \;\; {\mathbf n}^T_{y,n_e}]^T$. From
\eqref{EQ_Eve_MISOSE}, it is clear that when $\sigma_{{\mathbf h}} \geq
\sigma_{{\mathbf g}}$, we have the markov relationship that given ${\mathbf H}'$
\begin{equation} \label{eq_degraded_markov}
{\mathbf x} \rightarrow \frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}' \rightarrow
{\mathbf z}.
\end{equation}
The degraded MIMOME wiretap channel
$({\mathbf x},(\sigma_{{\mathbf g}}/\sigma_{{\mathbf h}}){\mathbf y}', {\mathbf z})$ is then formed.

Now based on the proposed degraded channel
$({\mathbf x},(\sigma_{{\mathbf g}}/\sigma_{{\mathbf h}}){\mathbf y}', {\mathbf z})$, we know that the
secrecy capacity is upper-bounded by
\begin{align}
C^{\;t}_s &\leq \underset{{\mathbf x}}{\max} \;
I({\mathbf x};\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}'|{\mathbf H}')-I({\mathbf x};{\mathbf z}|{\mathbf G}),
\label{eq_MIMOME_capacity_I_a} \\
&= \underset{{\mathbf x}}{\max} \; I({\mathbf x};{\mathbf y}|{\mathbf H})-I({\mathbf x};{\mathbf z}|{\mathbf G}),
\label{eq_MIMOME_capacity_I}
\end{align}
where the inequality follows \cite{Secrecy_sc}, with the
right-hand-side (RHS) being the secrecy capacity of our degraded
channel but forcing the eavesdropper knowing $({\mathbf z},{\mathbf G})$ instead
of $({\mathbf z},{\mathbf H}',{\mathbf G})$; and the equality comes from that
$({\mathbf y}',{\mathbf H}')$ and $({\mathbf y},{\mathbf H})$ have the same distributions. From
\cite{csiszar1978broadcast}, we also know that the RHS of
\eqref{eq_MIMOME_capacity_I} is achievable. Thus the RHS of
\eqref{eq_MIMOME_capacity_I} is the secrecy capacity $C^{\;t}_s$.
Furthermore, from \cite{Khisti_MIMOME}, we know that Gaussian
${\mathbf x}$ is optimal for the secrecy capacity in
\eqref{eq_MIMOME_capacity_I} under total power constraint
\eqref{power_cons}. Then our claim is valid.
\end{proof}

Note that on the contrary to the upper-bound with full CSIT in
\cite{Khisti_MIMOME}, our secrecy capacity upper bound
\eqref{eq_MIMOME_capacity_I_a} is independent of the realizations
of ${\mathbf H}$ and ${\mathbf G}$. This is why our bounds are tight for MIMOME
channels with only statistical CSIT. Compared to the proof for the
MISOSE secrecy capacity \cite{Secrecy_sc}, the key for deriving
the tight MIMOME upper-bound is separating the legitimate channel
matrix ${\mathbf H}$ by two submatrices ${\mathbf H}_{(n_r-n_e)}$ and ${\mathbf H}_{n_e}$
as \eqref{eq_split_H}, and only introducing correlations between
${\mathbf H}_{n_e}$ and ${\mathbf G}$ as \eqref{eq_H_divide}. One can treat the
unchanged submatrix ${\mathbf H}_{(n_r-n_e)}$ as a ``safe'' channel matrix
without being eavesdropped, which provides SNR scaling for the
secrecy capacity. In MISOSE channel, such a ${\mathbf H}_{(n_r-n_e)}$ does
not exist because $n_r=n_e=1$, and there is no SNR scaling. This
intuition is verified rigorously from the upcoming Theorem
\ref{Theorem_No_CQI} and Corollary \ref{coro_total_SNR}.

Now we fully characterize the MIMOME secrecy capacity based on
Lemma \ref{Lem_No_CQI}. Typically, the MIMOME secrecy capacity
optimization problems like the upcoming \eqref{eq_MISOSE_capacity}
are non-concave. This is due to that the MIMOME secrecy
capacities, such as \eqref{eq_MISOSE_capacity}, are a difference
of two concave functions. However, with aids of the degraded
MIMOME channels formed by \eqref{eq_H_divide}, the stochastic
MIMOME optimization problem \eqref{eq_MISOSE_capacity} can be
transformed to be a Schur-concave problem. This transformation
helps a lot to find the optimal solution
\eqref{eq_total_cons_optimal}. Note that the completely monotone
property for MISOSE optimization problem \cite{Secrecy_sc} may not
exist for the MIMOME one, and thus the method in \cite{Secrecy_sc}
is
hard to be extended to the MIMOME cases. \\
\begin{Theo} \label{Theorem_No_CQI}
Under the total power constraint
\begin{equation} \label{power_cons1}
\mathrm{Tr}\left(\Sigma_{{\mathbf x}}\right) \leq P,
\end{equation}
the MIMOME secrecy capacity $C^{\;t}_s$ when $n_r \geq n_e$ and
$\sigma_h \geq \sigma_g$ is
\begin{equation}
\max_{\Sigma_{{\mathbf x}}}\!\left({\mathds E}_{{\mathbf H}}\!\left[\!\log\left|{\mathbf I}+{\mathbf H}\Sigma_{{\mathbf x}}{\mathbf H}^\dag\!\right|\!\right]\!-\!{\mathds E}_{{\mathbf G}}\!\left[\!\log\!\left|{\mathbf I}+{\mathbf G}\Sigma_{{\mathbf x}}{\mathbf G}^\dag\!\right|\!\right]\!\right),
\label{eq_MISOSE_capacity}
\end{equation}
and the optimal channel input covariance matrix subject to
\eqref{power_cons1} is
\begin{equation} \label{eq_total_cons_optimal}
{\Sigma}^*_{{\mathbf x}}=\frac{P}{n_t}{\mathbf I}.
\end{equation}
\[
\]
\end{Theo}

\begin{proof}
First we show that under our setting, subject to
\eqref{power_cons1}, the stochastic MIMOME optimization problem
\eqref{eq_MISOSE_capacity} can be transformed to an equivalent
concave problem. The key is cleverly using the same marginal
channel formed by \eqref{eq_H_divide}. Note that the objective
function in \eqref{eq_MISOSE_capacity} can be rewritten as
$I({\mathbf x};{\mathbf y}|{\mathbf H})-I({\mathbf x};{\mathbf z}|{\mathbf G})$ with ${\mathbf x} \sim CN(0,\Sigma_{{\mathbf x}})$
as
\begin{align}
I({\mathbf x};{\mathbf y}|{\mathbf H})-I({\mathbf x};{\mathbf z}|{\mathbf G}) \overset{(a)}=&
I({\mathbf x};\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}'|{\mathbf H}')-I({\mathbf x};{\mathbf z}|{\mathbf G}),
\notag \\
\overset{(b)}=&
I({\mathbf x};\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}'|{\mathbf z},{\mathbf H}'),
\label{eq_Convex}
\end{align}
and then from \cite{Khisti_MIMOME}, we know that the RHS of
(\ref{eq_Convex} b) is concave in $\Sigma_{{\mathbf x}}$. The
(\ref{eq_Convex} a) comes from \eqref{eq_MIMOME_capacity_I}, while
(\ref{eq_Convex} b) comes from \eqref{eq_H_divide} and the fact
that
\begin{align}
&I({\mathbf x};\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}'|{\mathbf H}')-I({\mathbf x};{\mathbf z}|{\mathbf G})
\notag \\
\overset{(a)}=&
I({\mathbf x};\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}',{\mathbf z}|{\mathbf H}_{(n_r-n_e)},{\mathbf G})-I({\mathbf x};{\mathbf z}|{\mathbf H}_{(n_r-n_e)},{\mathbf G}),
\notag \\
\overset{(b)}=&
I({\mathbf x};\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}'|{\mathbf z},{\mathbf H}_{(n_r-n_e)},{\mathbf G}).
\label{eq_Convex1}
\end{align}
In the above, (\ref{eq_Convex1} a) comes from that
\begin{align}
&I({\mathbf x};\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}'|{\mathbf H}')=I({\mathbf x};\frac{\sigma_{{\mathbf g}}}{\sigma_{{\mathbf h}}}{\mathbf y}',{\mathbf z}|{\mathbf H}_{(n_r-n_e)},{\mathbf G}),
\notag \\ \mbox{and} \;\;\;
&I({\mathbf x};{\mathbf z}|{\mathbf G})=I({\mathbf x};{\mathbf z}|{\mathbf H}_{(n_r-n_e)},{\mathbf G}) \notag,
\end{align}
with the former resulting from \eqref{eq_H_divide} and the Markov
relationship \eqref{eq_degraded_markov} and the latter resulting
from the independence of ${\mathbf H}_{(n_r-n_e)}$ with $({\mathbf x},{\mathbf z},{\mathbf G})$;
while (\ref{eq_Convex1} b) comes from the chain rule of the mutual
information \cite{Book_Cover}.

After showing that \eqref{eq_MISOSE_capacity} can be transformed
to an equivalent concave problem, now we show that
\eqref{eq_MISOSE_capacity} can be further transformed to a
symmetric optimization problem. Then we can explore the
Schuar-concavity of the secrecy capacity
\eqref{eq_MISOSE_capacity} to show that the optimal
$\Sigma_{{\mathbf x}}=\alpha{\mathbf I}$ where $0 \leq \alpha \leq P/n_t$. For any
non-diagonal ${\Sigma}_{{\mathbf x}}$, we can apply the eigenvalue
decomposition on it as ${\Sigma}_{{\mathbf x}}={\mathbf U}{\mathbf D}{\mathbf U}^\dag$, where
${\mathbf U}$ is unitary and ${\mathbf D}$ is diagonal. Then for the objective
function in \eqref{eq_MISOSE_capacity}, setting
${\Sigma}_{{\mathbf x}}={\mathbf U}{\mathbf D}{\mathbf U}^\dag$ and ${\Sigma}_{{\mathbf x}}={\mathbf D}$ will
result in the same value since ${\mathbf H}$ and ${\mathbf G}$ are Rayleigh
distributed as \eqref{eq_Rayleigh}. Also the total power
constraint \eqref{power_cons1} can be transformed as
\begin{equation} \label{eq_evd_cons}
\mathrm{Tr}\left({\mathbf U}{\mathbf D}{\mathbf U}^\dag\right)=\mathrm{Tr}\left({\mathbf U}^\dag{\mathbf U}{\mathbf D}\right)=\mathrm{Tr}\left({\mathbf D}\right)
\leq P.
\end{equation}
In the following, we then focus on the following optimization
problem instead
\begin{equation}
\max_{{\mathbf D}}\!\left({\mathds E}_{{\mathbf H}}\!\left[\!\log\left|{\mathbf I}+{\mathbf H}{\mathbf D}{\mathbf H}^\dag\!\right|\!\right]\!-\!{\mathds E}_{{\mathbf G}}\!\left[\!\log\!\left|{\mathbf I}+{\mathbf G}{\mathbf D}{\mathbf G}^\dag\!\right|\!\right]\!\right),
\label{eq_MISOSE_capacityD}
\end{equation}
where the diagonal matrix ${\mathbf D}$ satisfying \eqref{eq_evd_cons}.
Because ${\mathbf H}$ and ${\mathbf G}$ are Rayleigh faded as \eqref{eq_Rayleigh},
\eqref{eq_MISOSE_capacityD} is symmetric for ${\mathbf D}$. That is, all
permutations of the diagonal terms of ${\mathbf D}$ will result in the
same value for the objective function in
\eqref{eq_MISOSE_capacityD}. Together with the fact that
\eqref{eq_MISOSE_capacityD} is concave as shown previously, we
know that the objective function \eqref{eq_MISOSE_capacityD} is
Schur-concave. Subject to the constraint $\mathrm{Tr}({\mathbf D}) \leq
P$, from \cite{Marshall_Inequalities}, it can be easily shown that
the optimal ${\mathbf D}$ (and thus $\Sigma_{{\mathbf x}}$) is $\alpha {\mathbf I}$ where
$0 \leq \alpha \leq P/n_t$. This result comes from the fact that
the diagonal entries $\{\alpha,\ldots,\alpha\}$ are majorized by
any other diagonal entries $\{d_1,\ldots,d_{n_t}\}$ of ${\mathbf D}$ if
$\frac{1}{n_t}\sum_{i=1}^{n_t}{d_i}=\alpha$, and properties of the
Schur-concave function \cite{Marshall_Inequalities}.

Finally, we will show that using all available power, that is,
$\Sigma_{{\mathbf x}}=(P/n_t){\mathbf I}$ is optimal for
\eqref{eq_MISOSE_capacity}. To do this, we will first show an
important property of the objective function in
\eqref{eq_MISOSE_capacity} with respect to the matrix partial
ordering as in the following Lemma.
\\

\begin{lemma} \label{lem_mono}
For the objective function of the optimization problem in
\eqref{eq_MISOSE_capacity},
\begin{equation} \label{eq_capacity_object}
R_s(\Sigma_{{\mathbf x}}) \triangleq
{\mathds E}_{{\mathbf H}}\!\left[\!\log\left|{\mathbf I}+{\mathbf H}\Sigma_{x}{\mathbf H}^\dag\!\right|\!\right]\!-\!{\mathds E}_{{\mathbf G}}\!\left[\!\log\!\left|{\mathbf I}+{\mathbf G}\Sigma_{x}{\mathbf G}^\dag\!\right|\!\right],
\end{equation}
we have the following properties when $n_r \geq n_e$ and
$\sigma_h \geq \sigma_g$ \\

\noindent (I) If two channel input covariance matrices satisfy
$\Sigma^2_x \succeq \Sigma^1_x \succeq \mathbf{0}$, then
\begin{equation} \label{eq_mono_sigma_x}
R_s(\Sigma^2_x)\geq R_s(\Sigma^1_x).
\end{equation} \\
(II) If $\Sigma^1_x \succ \mathbf{0}$, then $R_s(\Sigma^1_x)>0$.
\[
\]
\end{lemma}
The proof of Lemma \ref{lem_mono} is given in Appendix
\ref{app_mono}, which results from the equivalent degraded MIMOME
channels described in the proof of Lemma \ref{Lem_No_CQI}. As
aforementioned, we know the optimal $\Sigma_{{\mathbf x}}=\alpha {\mathbf I}$ for
\eqref{eq_MISOSE_capacity} subject to \eqref{power_cons1}. Now we
can show that $\alpha=P/n_t$ is optimal for
\eqref{eq_MISOSE_capacity} using Lemma \ref{lem_mono} as follows.
Let us substitute $\Sigma_{{\mathbf x}}=\alpha{\mathbf I}$ into the objective
function in \eqref{eq_MISOSE_capacity}, it becomes
\begin{equation} \label{eq_total_alpha}
{\mathds E}_{{\mathbf H}}\!\left[\!\log\left|{\mathbf I}+\alpha{\mathbf H}\bH^\dag\!\right|\!\right]\!-\!{\mathds E}_{{\mathbf G}}\!\left[\!\log\!\left|{\mathbf I}+\alpha{\mathbf G}\bG^\dag\!\right|\!\right]=C(\alpha),
\end{equation}
where $0 \leq \alpha \leq P/n_t$. If $\alpha>0$, we know that
\[
(P/n_t){\mathbf I} \succeq \alpha {\mathbf I} \succ \mathbf{0}.
\]
Then from Property (I) of Lemma \ref{lem_mono}, we know that
$C(P/n_t) \geq C(\alpha)$ for any $\alpha>0$. As for the case
$\alpha = 0$, we know that $C(0)=0$ from \eqref{eq_total_alpha}.
However, if $\alpha>0$, we have $ C(\alpha)>0$ from Property (II)
of Lemma \ref{lem_mono}. Then we know that
\[
C(P/n_t) \geq C(\alpha)> C(0),
\]
for any $\alpha>0$. Thus $\alpha=P/n_t$ is optimal for
\eqref{eq_total_alpha}. It concludes our proof.
\end{proof}

\textit{Remark:} Note that in the final step of Theorem
\ref{Theorem_No_CQI}'s proof, we show that the optimal channel
input covariance matrix happens when the total power constraint in
\eqref{power_cons1} is met with equality. This fact is not always
true for the wiretap channel and counter examples are given
\cite{Secrecy_Finite}\cite{Li_fading_secrecy_j}. When the transmit
power increases, both the SNRs at the legitimate receiver and the
eavesdropper increase. Then the secrecy capacity may not always be
maximized with using all available power. Indeed, this property
was also examined in \cite{Secrecy_scJSAC}. However, in
\cite{Secrecy_scJSAC}, the perfect CSIT of the legitimate receiver
is assumed to be  available, which is fundamentally different to
our setting.

Now we have the following result which characterizes the secrecy
capacity in Theorem \ref{Theorem_No_CQI} with respect to the
number of antennas $(n_t,n_r,n_e)$ at the transmitter, legitimate
receiver and eavesdropper respectively. The proof is given in
Appendix
\ref{App_coro_total_SNR}. \\

\begin{coro} \label{coro_total_SNR}
Under total power constraint \eqref{power_cons1}, we have the
following asymptotic results for the MIMOME secrecy capacity
$C^{\;t}_s$ as
\[
\lim_{P_g \rightarrow
\infty}\frac{C^{\;t}_s-n_r\log(\sigma^2_h/\sigma^2_g)}{\log
P_g}=\left\{\begin{array}{lc} n_r-n_e, & n_t \geq n_r > n_e\\
n_t-n_e, & n_r \geq n_t > n_e\\
0, &   n_r \geq n_e \geq n_t\\
\end{array}\right.
\]
when $\sigma_h \geq \sigma_g > 0$, where $P_g=\sigma^2_gP$ is the
equivalent received SNR at the eavesdropper. \\
\end{coro}

In \cite{Secrecy_sc}, it was shown that when $n_r=n_e=1$, the
secrecy capacity does not scale with $P$. Our Corollary
\ref{coro_total_SNR} shows that to make secrecy capacity scale
with $P$, we must let $n_r-n_e$ larger than (but not equal to)
zero. Also adding enough number of transmit antennas to make
$n_t>n_e$ is very important for increasing the secrecy capacity.
Indeed, from our numerical results in Section
\ref{Sec_simulation}, with fixed $n_t>n_e$, increasing the
difference $n_r-n_e$ will help to increase the secrecy capacities
for all SNR regimes (besides the high SNR regime proved in
Corollary \ref{coro_total_SNR}).

From Property (II) of Lemma \ref{lem_mono}, the secrecy capacity
in Theorem \ref{Theorem_No_CQI} is always positive when $P>0$,
$n_r \geq n_e$ and $\sigma_h > \sigma_g$. It will also be
interesting to see when the secrecy capacity is zero. We have the
following result, where the proof is similar
to that of Lemma \ref{Lem_No_CQI} and neglected.\\

\begin{coro}
\label{Coro_No_CQI_zero} For the MIMOME fast Rayleigh fading
wiretap channel \eqref{EQ_main_MISOSE}\eqref{EQ_Eve_MISOSE} with
the statistical CSIT of $\mathbf{H}$ and $\mathbf{G}$, the secrecy
capacity is zero when $n_r \leq n_e$ and $\sigma_h \leq \sigma_g$.
\end{coro}

\section{Secrecy capacity under the \\ per-antenna power constraint}\label{Sec_per_antenna}

After investigating the MIMOME secrecy capacity $C^{\;t}_s$ under
the total power constraint over all transmitter antennas, now we
consider the MIMOME channel under the more practical per-antenna
power constraint. First, we show that the MIMOME secrecy capacity
under the per-antenna power constraint can be expressed as the
following convex optimization problem, with proof given in
Appendix
\ref{app_Pro_MIMO_per_antenna}. \\

\begin{proposition} \label{Pro_MIMO_per_antenna}
Under the per-antenna power constraint
\begin{equation} \label{eq_per_power_cons}
\{\Sigma_{{\mathbf x}}\}_{ii} \leq P_i,
\end{equation}
$i=1,\ldots,n_t$, when $n_r \geq n_e$ and $\sigma_h \geq
\sigma_g>0$, the MIMOME secrecy capacity $C^p_s$ is
\begin{equation} \label{eq_per_MIMOME_capacity_old}
\max_{\Sigma_{{\mathbf x}}}\!\left({\mathds E}_{{\mathbf H}}\!\left[\!\log\left|{\mathbf I}+{\mathbf H}\Sigma_{{\mathbf x}}{\mathbf H}^\dag\!\right|\!\right]\!-\!{\mathds E}_{{\mathbf G}}\!\left[\!\log\!\left|{\mathbf I}+{\mathbf G}\Sigma_{{\mathbf x}}{\mathbf G}^\dag\!\right|\!\right]\!\right),
\end{equation}
which can be transformed to a concave optimization problem subject
to \eqref{eq_per_power_cons} as
\begin{align}
\max_{\Sigma_{{\mathbf x}}}\Bigg(&{\mathds E}_{{\mathbf G},{\mathbf H}_{(n_r-n_e)}}\!\left[\!\log\left|{\mathbf I}+\Sigma_{{\mathbf x}}\left({\mathbf H}^\dag_{(n_r-n_e)}{\mathbf H}_{(n_r-n_e)}+\frac{\sigma^2_h}{\sigma^2_g}{\mathbf G}^\dag{\mathbf G}\right)\!\right|\right]
\notag
\\&-\!{\mathds E}_{{\mathbf G}}\!\left[\!\log\!\left|{\mathbf I}+{\mathbf G}\Sigma_{{\mathbf x}}{\mathbf G}^\dag\!\right|\!\right]
\Bigg),
 \label{eq_per_MIMOME_capacity}
\end{align}
where ${\mathbf G}$ is the eavesdropper channel matrix and
${\mathbf H}_{(n_r-n_e)}$ is the sub-matrix of the legitimate channel as
defined in \eqref{eq_H_divide}. \\
\end{proposition}

Unlike the optimization in Theorem \ref{Theorem_No_CQI}, subject
to \eqref{eq_per_power_cons}, the optimal $\Sigma_{{\mathbf x}}$ of
problem \eqref{eq_per_MIMOME_capacity_old} is very hard to find
analytically for general $n_r$. The main difficulty is that the
per-antenna power constraint \eqref{eq_per_power_cons} is not
symmetric. Note that \eqref{eq_per_power_cons} is related to
eigenvectors of $\Sigma_{\mathbf x}$, while the total power constraint
\eqref{power_cons1} ($\mathrm{Tr}\{\Sigma_{\mathbf x}\}$) is independent
of these eigenvectors. Thus although the total power constraint
can be transformed to \eqref{eq_evd_cons} via the eigenvalue
decomposition of $\Sigma_{{\mathbf x}}$, the same technique cannot be
applied to the per-antenna constraint \eqref{power_cons1}. Thus we
only claim that the problem \eqref{eq_per_MIMOME_capacity_old}
subject to \eqref{eq_per_power_cons} can be transformed to a
concave one as \eqref{eq_per_MIMOME_capacity}, but not the
Schuar-concave one as in the proof of Theorem
\ref{Theorem_No_CQI}. Lack of symmetry in constraint
\eqref{eq_per_power_cons} makes finding analytical solution of it
difficult. Nevertheless, we still have the following result for
the structure of the optimal
$\Sigma^*_x$ in Proposition \ref{Pro_MIMO_per_antenna} as \\

\begin{proposition}
\label{Pro_Full_power} Under the per-antenna power constraint
$\{\Sigma_{{\mathbf x}}\}_{ii} \leq P_i, i=1,\ldots,n_t$, when $n_r \geq
n_e$ and $\sigma_h \geq \sigma_g$, there exists an optimal channel
input covariance matrix $\Sigma^*_x$ for the MIMOME secrecy
capacity optimization problem \eqref{eq_per_MIMOME_capacity_old}
which satisfies
\begin{equation} \label{eq_per_equality}
\{\Sigma^*_{{\mathbf x}}\}_{ii}=P_i, \;\;\; i=1,\ldots,n_t,
\end{equation}
that is, each antenna should fully use its available power. \\
\end{proposition}

\begin{proof}
We will prove it by showing that if there exists an optimal
$\Sigma_{{\mathbf x}}$ which does not meet \eqref{eq_per_equality}, one
can find another $\Sigma^*_{\mathbf x}$ meeting \eqref{eq_per_equality}
and having the same optimal value. Suppose that there exists an
optimal $\Sigma_{{\mathbf x}}$ maximizing
\eqref{eq_per_MIMOME_capacity_old} which has some diagonal terms
using powers smaller than their maximum available powers. We
collect the indexes corresponding to these diagonal terms as a
non-empty set $\mathds{I}_{s} \neq \varnothing$, i.e.,
$\{\Sigma_{{\mathbf x}}\}_{ii}<P_i,$ if $ i \in \mathcal{I}_{s} \subseteq
\{1,\ldots, n_t\}$. Then we can form another $\Sigma^*_{\mathbf x}$ as
\[
\{\Sigma^*_{{\mathbf x}}\}_{ij}=\left\{
\begin{array}{ll}
P_i & \mbox{if $i=j$ and $i \in \mathds{I}_{s} $,} \\
\{\Sigma_{{\mathbf x}}\}_{ij} & \mbox{otherwise.}
\end{array} \right.
\]
Then we know that $\Sigma^*_{{\mathbf x}}-\Sigma_{{\mathbf x}} \succ 0$, because
$\Sigma^*_{{\mathbf x}}-\Sigma_{{\mathbf x}}$ is a diagonal matrix with
$|\mathds{I}_{s}|$ positive diagonal terms, while the other
diagonal terms are zero. Note that the size of set
$\mathds{I}_{s}$ meets $|\mathds{I}_{s}|>0$ and at least one
diagonal terms of $\Sigma^*_{{\mathbf x}}-\Sigma_{{\mathbf x}}$ is positive. Then
from Property (I) of Lemma \ref{lem_mono} we know that for the
objective function $R_s(.)$ in \eqref{eq_per_MIMOME_capacity_old},
$R_s(\Sigma^*_{{\mathbf x}}) \geq R_s(\Sigma_{{\mathbf x}})$. Since $\Sigma_{{\mathbf x}}$
is assumed to be optimal, $R_s(\Sigma_{{\mathbf x}}) \geq
R_s(\Sigma^*_{{\mathbf x}})$, and then we know that $R_s(\Sigma^*_{{\mathbf x}}) =
R_s(\Sigma_{{\mathbf x}})$.
\end{proof}

Although for general $n_r$, we only can obtain the above property
for the optimization problem in Proposition \ref{Lem_No_CQI}, due
to the aforementioned difficulty below Proposition
\ref{Lem_No_CQI}. For the special MISOSE case $n_r=n_e=1$, we can
have the following secrecy capacity result as in the upcoming
Theorem. The key proof steps come as follows. First, we use
\eqref{eq_per_MIMOME_capacity} in Proposition
\ref{Pro_MIMO_per_antenna} to transform the objective function in
\eqref{eq_per_MIMOME_capacity_old} into a simpler one as in the
upcoming \eqref{eq_Cs_optimizae}. And then we can apply
Proposition \ref{Pro_Full_power} and properties of
random Gaussian vectors to obtain analytical solutions. \\
\begin{Theo}\label{Theorem_Per}\it
Subject to the per-antenna power constraints $
\{\Sigma_{{\mathbf x}}\}_{ii} \leq P_i, $ $i=1,\ldots,n_t$, the secrecy
capacity optimization problem \eqref{eq_per_MIMOME_capacity} has
optimal channel input covariance matrix
\begin{equation} \label{eq_per_optimal}
\Sigma^*_{{\mathbf x}}=\mathrm{diag}\left(P_1,P_2,\ldots,P_{n_t}\right)
\end{equation}
for the MISOSE wiretap channel, where the legitimate receiver and
eavesdropper has single antenna each $n_r=n_e=1$. \\
\end{Theo}

\begin{proof}
For this special case $n_r=n_e=1$, the sub-matrix of the
legitimate channel ${\mathbf H}_{(n_r-n_e)}$ in
\eqref{eq_per_MIMOME_capacity} does not exist. Then the
optimization problem in \eqref{eq_per_MIMOME_capacity} can be
rewritten as
\begin{align}
&\max_{\Sigma_{{\mathbf x}}}\!\!\left(\!\!\!{\mathds E}_{\mathbf{g}}\!\left[\log\frac{\sigma_{\mathbf{g}}^2/\sigma_{\mathbf{h}}^2+\mathbf{g}^\dag\Sigma_{\mathbf{x}}\mathbf{g}}{\sigma_{\mathbf{g}}^2/\sigma_{\mathbf{h}}^2}\right]\!-\!{\mathds E}_{\mathbf{g}}\!\left[\log(1+\mathbf{g}^\dag\Sigma_{\mathbf{x}}\mathbf{g})\right]\!\!\!\right),
\notag \\
=&\max_{\Sigma_{{\mathbf x}}}\!\!\left(\!\!\!{\mathds E}_{\mathbf{g}}\!\left[\log\left(1+\frac{\sigma_{\mathbf{g}}^2/\sigma_{\mathbf{h}}^2-1}{1+\mathbf{g}^\dag\Sigma_{\mathbf{x}}\mathbf{g}}\right)\right]\right)+\!\log\left(\sigma_{\mathbf{h}}^2/\sigma_{\mathbf{g}}^2\right),
\label{eq_Cs_optimizae}
\end{align}
where for clearness, we replace the notation ${\mathbf G}$ in
\eqref{eq_per_MIMOME_capacity} with $\mathbf{g}^\dag\sim CN(0,
\sigma_{{\mathbf g}}^2 \mathbf{I})$ to emphasize that the eavesdropper
channel is now a vector when $n_r=1$. Note that the objective
function in \eqref{eq_Cs_optimizae} is only related to the channel
vector ${\mathbf g}$ for the eavesdropper, but not the channel vector for
the legitimate receiver. The above transformations make finding
analytical solutions possible.

Next, from Proposition \ref{Pro_Full_power}, without loss of
optimality, one can replace the inequality constraint
\eqref{eq_per_power_cons} by the equality constraint on the
diagonal entries of $\Sigma_{{\mathbf x}}$ as
\begin{equation} \label{eq_per_equality_cons}
\left\{\Sigma_{{\mathbf x}}\right\}_{ii}= P_i,
\end{equation} for
$i=1,\ldots, n_t$. Now we show that for any $\beta \leq 0$,
$\sigma > 0$, $\beta +2\sigma > 0$, subject to equality constraint
\eqref{eq_per_equality_cons}, the optimization problem
\begin{equation} \label{eq_per_MISOSE_Cs_optimizae}
\max_{\Sigma_{{\mathbf x}}}\!\!\left(\!\!\!{\mathds E}_{\mathbf{g}}\!\left[\log\left(1+\frac{\beta}{\sigma+\mathbf{g}^\dag\Sigma_{\mathbf{x}}\mathbf{g}}\right)\right]\right)
\end{equation}
has optimal solution \eqref{eq_per_optimal}. This fact is proved
in Appendix \ref{App_per_MISOSE_capcity}, where the key step is
cleverly applying properties of random Gaussian vectors. Note that
the objective function in \eqref{eq_Cs_optimizae} is a special
case of that in \eqref{eq_per_MISOSE_Cs_optimizae} with
$\beta=\sigma_{\mathbf{g}}^2/\sigma_{\mathbf{h}}^2-1$ and
$\sigma=1$. And then subject to \eqref{eq_per_power_cons}, the
optimal $\Sigma_{{\mathbf x}}$ maximizing \eqref{eq_per_MIMOME_capacity}
(and the the original \eqref{eq_per_MIMOME_capacity_old}) with
$n_r=n_e=1$ is \eqref{eq_per_optimal}. Our claim in Theorem
\ref{Theorem_Per} is valid.
\end{proof}

Here we investigate the SNR scaling of the MISOSE channel under
the per-antenna power constrains. From Theorem \ref{Theorem_Per},
we have
\begin{coro} \label{coro_per_SNR_MISOSE}
Under the per-antenna power constraint $ \{\Sigma_{{\mathbf x}}\}_{ii}
\leq P_i, $ $i=1,\ldots,n_t$, when $n_r=n_e=1$ and $\sigma_h \geq
\sigma_g > 0$, we have the following asymptotic results for the
MISOSE secrecy capacity $C^p_s$ as
\begin{equation} \label{eq_per_DoF_MISOSE}
\lim_{P^g_{max} \rightarrow
\infty}\frac{C^p_s-\log(\sigma^2_h/\sigma^2_g)}{\log P^g_{max}}=0,
\end{equation}
where $P^g_{max}=\sigma^2_g P_{max}$ with
\begin{equation} \label{eq_per_Pmax}
P_{max} \triangleq \max_{i=1,\ldots n_t}P_i
\end{equation}
being the maximum among per-antenna power constraints. \\
\end{coro}

\begin{proof}
As \eqref{eq_capacity_object}, let us denote the objective
function of optimization problem
\eqref{eq_per_MIMOME_capacity_old} as $R_s(\Sigma_{{\mathbf x}})$. From
the Property (I) of Lemma \ref{lem_mono}, we know that the MISOSE
secrecy capacity under the per-antenna power constraint
$C^p_s=R_s(diag\{P_1,\ldots,P_{n_t}\}) \leq
R_s(diag\{P_{max},\ldots,P_{max}\})$. This fact is due to that
\[
diag\{P_{max},\ldots,P_{max}\}\succeq
diag\{P_1,\ldots,P_{n_t}\}\succeq \mathbf{0},
\]
because
$diag\{P_{max},\ldots,P_{max}\}-diag\{P_1,\ldots,P_{n_t}\}$ is a
diagonal matrix with non-negative entries from
\eqref{eq_per_Pmax}. By Theorem \ref{Theorem_No_CQI} we know that
$R_s(diag\{P_{max},\ldots,P_{max}\})$ is the MISOSE secrecy
capacity under total power constraint $\mathrm{Tr}(\Sigma_{{\mathbf x}})
\leq n_tP_{max}$. Then by Corollary \ref{coro_total_SNR}, for
fixed $n_t$, \eqref{eq_per_DoF_MISOSE} is valid since $n_r=n_e=1$.
\end{proof}

From Corollary \ref{coro_per_SNR_MISOSE}, we know that the
multiple transmitter antennas for the MISOSE channel limitedly
help to increase the secrecy capacity under the per-antenna power
constraint. Indeed, note that \eqref{eq_per_DoF_MISOSE} is hold
even when only one transmitter antenna, which has the maximum
power constraint, is selected to transmit secret messages while
the other ones are silent. In this case, the per-antenna power
constraint corresponding to \eqref{eq_per_power_cons} are
$P_i=P_{max}$ and $P_j=0, j \neq i, \forall j$, where the $i$th
transmitter antenna is selected. When SNR is large enough, the
secrecy rate for this simple transmitter antenna selection scheme
\cite{molisch2004mimo} is within constant gap compared with the
secrecy capacity where all transmitter antennas are used.

To overcome the negative results for MISOSE channels revealed in
Corollary \ref{coro_per_SNR_MISOSE}, as in Section
\ref{Sec_total_power}, we now show that making the difference
between numbers of antennas at the legitimate receiver and
eavesdropper $n_r-n_e>0$ will help to increase the secrecy
capacity under the per-antenna power constraint. Although we
cannot find the optimal $\Sigma_{{\mathbf x}}$ for
\eqref{eq_per_MIMOME_capacity_old} subject to
\eqref{eq_per_power_cons}, we use the following sub-optimal
$\Sigma_{{\mathbf x}}$ to compute the secrecy capacity lower bound as
$\Sigma_{{\mathbf x}}=\mathrm{diag}\left(P_1,P_2,\ldots,P_{n_t}\right)$.
This selection of $\Sigma_{{\mathbf x}}$ is optimal for the MISOSE case
from Theorem \ref{Theorem_Per}, and for the MIMOME case, this
selection is also reasonable due to Proposition
\ref{Pro_Full_power}. Now we have the following result for the
secrecy capacity in Proposition \ref{Pro_MIMO_per_antenna} as
\begin{coro} \label{coro_per_SNR}
Under the per-antenna power constraint $ \{\Sigma_{{\mathbf x}}\}_{ii}
\leq P_i, $ $i=1,\ldots,n_t$, when $\sigma_h \geq \sigma_g>0$, we
have the following asymptotic result for the MIMOME secrecy
capacity $C^p_s$ as
\begin{equation} \label{eq_per_DoF_MIMOME}
\lim_{P^g_{min} \rightarrow
\infty}\frac{C^p_s-n_r\log(\sigma^2_h/\sigma^2_g)}{\log
P^g_{min}} \geq \left\{\begin{array}{lc} n_r-n_e, & n_t \geq n_r > n_e\\
n_t-n_e, & n_r \geq n_t > n_e,
\end{array}\right.
\end{equation}
where $P^g_{min}=\sigma^2_g P_{min}$ with
\begin{equation}\label{eq_Pmin}
P_{min} \triangleq \min_{i=1,\ldots n_t}P_i
\end{equation}
is the minimum among per-antenna power constraints.
\end{coro}

\begin{proof}
From the definition of $P_{min}$ in \eqref{eq_Pmin}, we know that
\[
diag\{P_1,\ldots,P_{n_t}\} \succeq diag\{P_{min},\ldots,P_{min}\}
\succ \mathbf{0}.
\]
Then we know that $C^p_s \geq R_s(diag\{P_1,\ldots,P_{n_t}\}) \geq
R_s(diag\{P_{min},\ldots,P_{min}\})$, where the second inequality
comes from Property (I) of Lemma \ref{lem_mono}. By Theorem
\ref{Theorem_No_CQI} we know that
$R_s(diag\{P_{min},\ldots,P_{min}\})$ is the MIMOME secrecy
capacity under total power constraint $\mathrm{Tr}(\Sigma_{{\mathbf x}})
\leq n_tP_{min}$. Then by Corollary \ref{coro_total_SNR}, for
fixed $n_t$, \eqref{eq_per_DoF_MIMOME} is valid.
\end{proof}

From \eqref{eq_per_DoF_MIMOME}, as discussions under Corollary
\ref{coro_total_SNR}, adding enough number of legitimate-receiver
antennas $n_r$ (also enough number of transmit antennas $n_t$) is
very important for increasing the secrecy capacity $C^p_s$.
Moreover, \eqref{eq_per_DoF_MIMOME} also reveals some antenna
selection rules for MIMOME wiretap channel under the per-antenna
power constraint. For example, when $n_t
> n_r
> n_e$, the transmitter can select the largest
$n_r$ antennas to transmit the secret message. Each selected
antenna transmits with all its allowable power. When the allowable
power of all selected antenna is high enough, this simple transmit
antenna selection scheme can achieve good secrecy rate performance
according to \eqref{eq_per_DoF_MIMOME}.

Finally, we develop algorithms to solve the MIMOME secrecy
capacity optimization problems \eqref{eq_per_MIMOME_capacity_old}
subject to the per-antenna power constraint
\eqref{eq_per_power_cons}. To do this, as in Proposition
\ref{Pro_MIMO_per_antenna}, by using the same marginal channel
matrix ${\mathbf H}'$ in \eqref{eq_H_divide}, we first transfer the
stochastic optimization problem \eqref{eq_per_MIMOME_capacity_old}
into a concave one \eqref{eq_per_MIMOME_capacity}. Next, from
Proposition \ref{Pro_Full_power}, one can set the inequalities in
constraints \eqref{eq_per_power_cons} with equalities as
\eqref{eq_per_equality} to further simplify the problem. Then we
can use algorithms similar to those in \cite{huh2010multiuser} to
find the optimal values for \eqref{eq_per_MIMOME_capacity_old} by
matrix calculus
\cite{hjorungnes2011complex}\cite{payaro2009hessian}. The gap to
the optimal value is within $n_t/t$, where $n_t$ is the number of
transmit antenna and $t>0$ is a parameter to prevent the algorithm
approaching non-semidefine $\Sigma_{{\mathbf x}}$. Note that our problem
can not be simplified by the
multiple-access-channels-broadcast-channels duality as
\cite{huh2010multiuser}. Though wiretap channels are similar to
the broadcast channels, there may be no corresponding dualities
for the wiretap channels.

\begin{figure*}[t]
\begin{align}
\nabla_x
f_t={\mathds E}_{\mathbf{H}'}\bigg[\big((\textbf{H}^{'})^\dagger\otimes(\textbf{H}^{'})^T\big)vec&\bigg(\big((\textbf{I}+\textbf{H}^{'}\Sigma_{{\mathbf x}}(\textbf{H}^{'})^\dagger)^{-1}\big)^T\bigg)-\big({\mathbf G}^\dagger\otimes{\mathbf G}^T\big)vec\bigg(\big((\textbf{I}+{\mathbf G}\Sigma_{{\mathbf x}}{\mathbf G}^\dagger)^{-1}\big)^T\bigg)\bigg]+\frac{1}{t}vec\left(\left({\Sigma_{{\mathbf x}}}^{-1}\right)^T\right)\label{eq_graft}
\end{align}

\begin{align}
\nabla^2_{xx}f_t
&={\mathds E}_{\mathbf{H}'}~\bigg[-((\textbf{H}^{'})^\dagger\otimes(\textbf{H}^{'})^T)\bigg(\big(\textbf{I}+\textbf{H}^{'}\Sigma_{{\mathbf x}}(\textbf{H}^{'})^\dagger)\otimes\big((\textbf{I}+\textbf{H}^{'}\Sigma_{{\mathbf x}}(\textbf{H}^{'})^\dagger)^{T}\bigg)^{-1}\cdot
\mathbf{K}_p
\bigg(((\textbf{H}^{'})^\dagger)^T\otimes\textbf{H}^{'}\bigg) \nonumber \\
&~~~~~~~~~~~+({\mathbf G}^\dagger\otimes{\mathbf G}^T)\bigg(\big(\textbf{I}+{\mathbf G}\Sigma_{{\mathbf x}}{\mathbf G}^\dagger)\otimes\big(\textbf{I}+{\mathbf G}\Sigma_{{\mathbf x}}{\mathbf G}^\dagger)^{T}\bigg)^{-1}\cdot
\mathbf{K}_p
\bigg(({\mathbf G}^\dagger)^T\otimes{\mathbf G}\bigg)\bigg]-\frac{1}{t}(\Sigma_{{\mathbf x}}\otimes\Sigma_{{\mathbf x}}^T)^{-1}\mathbf{K}_p
\label{eq_gragraft}
\end{align}

\hrulefill
\end{figure*}

Here comes the reformulation of the secrecy capacity optimization
problem \eqref{eq_per_MIMOME_capacity_old} subject to
\eqref{eq_per_power_cons} for our algorithm. First, we use the
following objective function as
\begin{equation} \label{eq_barrieried_objective}
f_t(\Sigma_{{\mathbf x}})=\tilde{R}_s(\Sigma_{{\mathbf x}})+\frac{1}{t}\log|\Sigma_{{\mathbf x}}|,
\end{equation}
where $\tilde{R}_s(.)$ is the concave objective function in
\eqref{eq_per_MIMOME_capacity} and $\frac{1}{t}\log|\Sigma_{{\mathbf x}}|$
is the logarithmic barrier. As for constraint
\eqref{eq_per_equality} (per-antenna power constraint
\eqref{eq_per_power_cons} with equality), we also rewrite it as
the following equality constraint
\begin{equation} \label{eq_Alg_constraint}
\mathbf{A}\cdot \emph{vec}(\Sigma_{{\mathbf x}}) =
[~P_1~\ldots~P_{n_t}]^T,
\end{equation}
where the $n_t \times (n_t)^2$ matrix $\mathbf{A}$ has entries
being $1$ for those corresponds to $\{\Sigma_{{\mathbf x}}\}_{ii}, i=1
\ldots n_t$, and entries being $0$ else, and the vectorization
operator for a matrix $\emph{vec}(.)$ is defined as
\cite{hjorungnes2011complex}\cite{payaro2009hessian}. For example,
when $n_t=2$
\[\mathbf{A}=
\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 0 & 0 & 1
\end{array}\right].
\]

Now we focus on the detailed steps for the optimization of
$f_t(\Sigma_{{\mathbf x}})$ in \eqref{eq_barrieried_objective} subject to
equality constraint \eqref{eq_Alg_constraint}. In the following,
the $\nu$ is the $n_t \times 1$ Lagrange multiplier associated to
equality constraint \eqref{eq_Alg_constraint}.

\begin{itemize}

\item[1)] Initialize $\Sigma^{(0)}_{{\mathbf x}}$ and $\nu^{(0)}$

\item[2)] By invoking the matrix calculus
\cite{hjorungnes2011complex}\cite{payaro2009hessian}, compute the
residue as
\[
{\mathbf r}= \left[\begin{array}{c}
\nabla_xf_t+{\textbf{A}}^T\nu\\
\textbf{A}\cdot \emph{vec}(\Sigma_{{\mathbf x}})-[~P_1~\ldots~P_{n_t}]^T
\end{array}\right]
\]
where $\nabla_x$ means the gradient of function respect to vector
$\emph{vec}(\Sigma_{{\mathbf x}})$, and $\nabla_xf_t$ is given in
\eqref{eq_graft} with ${\mathbf H}'$ from \eqref{eq_H_divide} and
$\otimes$ being the Kronecker product. The direction
$[\Delta_{x},\Delta_{\nu}]^T$ to update
$[\emph{vec}(\Sigma_{{\mathbf x}}),\nu]^T$ is computed by the KKT matrix
\cite{huh2010multiuser} as
\[
\left[\begin{array}{c}
\Delta_{x}\\
\Delta_\nu
\end{array}\right]
=- \left[\begin{array}{cc}
\nabla_{xx}^2f_t & \textbf{A}^T\\
\textbf{A} & 0
\end{array}\right]^{-1} {\mathbf r},
\]
where $\nabla_{xx}^2f_t$ is explicitly given in
\eqref{eq_gragraft}, with $\mathbf{K}_p$ being the permutation
matrix such that $vec(\mathbf{A}^T ) = \mathbf{K}_p
vec(\mathbf{A})$ as defined in \cite[(51)]{payaro2009hessian}.

\item[3)] For the $n+1$ step, we compute
$\left(\emph{vec}\left(\Sigma_{{\mathbf x}}^{(n+1)}\right),\nu^{(n+1)}\right)
=
\left(\emph{vec}\left(\Sigma_{{\mathbf x}}^{(n)}\right)+s\Delta_x,\nu^{(n)}+s\Delta\nu\right)$,
where $s$ is the step size found by the backtracking line search.

\item[4)] If the norm of residue
$\|{\mathbf r}\|_{_2}<\varepsilon$,~increase $t$ by a factor $\gamma=1.5$
and go to step 5. If not,~go back to step 2.

\item[5)] Stop when the gap $\frac{n_t}{t}<\varepsilon$,~if
not,~go back to step 2.
\end{itemize}

Note that steps 2 to 4 are the infeasible start Newton steps, and
the algorithm will converge since objective function $f_t(.)$ in
\eqref{eq_barrieried_objective} is concave (both $\tilde{R}_s(.)$
from \eqref{eq_per_MIMOME_capacity} and the logarithmic barrier
are concave) \cite{huh2010multiuser}. And following
\cite{huh2010multiuser}, one can show that the gap to the optimal
value of \eqref{eq_per_MIMOME_capacity_old} is $\frac{n_t}{t}$ as
in step 5, because the dimension of $\Sigma_{{\mathbf x}}$ in the
logarithmic barrier $\frac{1}{t}\log|\Sigma_{{\mathbf x}}|$ is $n_t \times
n_t$.

Compared with the algorithm in \cite{li2010optimal} which also
considers the per-antenna power constraint, our algorithm can
guarantee that the gap to the optimal value is vanishing when $t
\rightarrow \infty$ while that in \cite{li2010optimal} can not
guarantee such an optimality. Moreover, \cite{li2010optimal}
requires full CSIT of both legitimate channel and eavesdropper
while our statistical CSIT requirement is more practical.

\section{Simulations}\label{Sec_simulation}
In this section, we provide numerical and simulation results for
our theoretical claims in previous sections. In all figures
presented, noise at the legitimate receiver and eavesdropper has
unit variance each. The SNR is then defined as the total
transmitted power over all antennas in the dB scale. The ratio of
the legitimate channel variance over the eavesdropper channel
variance in \eqref{eq_Rayleigh} is $\sigma^2_h/\sigma^2_g=4$. All
channel matrices are Rayleigh faded.

First, in Fig. \ref{fig_capacity_different_P}-\ref{fig_alpha}, we
show the numerical results for secrecy capacities $C^{\;t}_s$
under the total power constraints \eqref{power_cons1} in Theorem
\ref{Theorem_No_CQI}. In Fig. \ref{fig_capacity_different_P}, we
compare the MIMOME secrecy capacities $C^{\;t}_s$ under different
combinations of number of antennas. Three different combinations
$(n_t,n_r,n_e)$ with fixed $n_t=4$ and $n_t \geq n_r$ are
considered, where $n_t,n_r$ and $n_e$ respectively are the number
of antennas at the transmitter, the legitimate receiver, and the
eavesdropper. Consistent with \cite{Secrecy_sc}, when $n_r=n_e=1$,
the MISOSE secrecy capacities do not scale with SNR and converges
at high SNR. Same phenomenon also happens even when $n_r=n_e=2$.
Thus it may be a waste of resource by increasing the SNR for the
MIMOME channel with $n_r=n_e$. To overcome this drawback, one can
increase $n_r$ to make it larger than $n_e$. These observations
meet our results in Corollary \ref{coro_total_SNR}. Also with
fixed number of transmit antennas $n_t>n_e$, increasing the
difference $n_r-n_e$ is very helpful to increase the secrecy
capacities in all SNR regimes. Similarly, as in Fig.
\ref{fig_capacity_different_P}, in Fig.
\ref{fig_capacity_different_n_t} we compare the MIMOME secrecy
capacities $C^{\;t}_s$ in Theorem \ref{Theorem_No_CQI} but with
fixed $n_r=5$ and $n_r \geq n_t$. The MISOSE secrecy capacities
with $(n_t,n_r,n_e)=(5,1,1)$ are also depicted in Fig
\ref{fig_capacity_different_n_t}, and again they do not scale with
the SNR. The SNR scaling in MIMOME channels can be obtained when
$n_r
> n_e$ but not $n_r=n_e$. And with fixed $n_r>n_e$ (but not $n_r=n_e$), increasing the
difference $n_t-n_e$ is very helpful to increase the secrecy
capacities even at medium SNR regimes.

Next, we consider the MIMOME secrecy capacities $C^{\;t}_s$ in
Theorem \ref{Theorem_No_CQI} with single transmit antenna in Fig.
\ref{fig_capacity_different_n_e}. As predicted in Corollary
\ref{coro_total_SNR}, when $n_t=1$ there will be no SNR scaling no
matter $n_r>n_e$ or $n_r=n_e$. However, one can observe that
channels with larger $n_r-n_e$ (but not larger $n_r$) will have
higher secrecy capacities $C^{\;t}_s$. Next, in Fig.
\ref{fig_alpha}, we show that all available transmit power $P$
should be used for MIMOME channels with total power constraints
\eqref{power_cons1}. We plot the $R_s(\alpha \mathbf{I})$s in
\eqref{eq_capacity_object}, the secrecy rates with channel input
matrices $\Sigma_{{\mathbf x}}=\alpha {\mathbf I}$. The maximum allowable transmit
power $P=20$ dB and $0 \leq \alpha \leq P/n_t$. From
Fig.\ref{fig_alpha}, the secrecy rate $R_s(\alpha \mathbf{I})$
increases monotonically with $\alpha$. Thus when $\alpha$ equals
to its maximum $P/n_t$, the secrecy rate $R_s(P/n_t \mathbf{I})$
is the secrecy capacity, as in the proof of Theorem
\ref{Theorem_No_CQI}.

\begin{figure} [t]
\centering
\epsfig{file=./Fig/fig_capacity_different_P.eps,width=0.5\textwidth}
\caption{Under total power constraints over all transmit antennas
\eqref{power_cons1}, the secrecy capacities versus SNRs with
different number of antennas $(n_t \geq n_r)$.}
\label{fig_capacity_different_P}
\end{figure}
\begin{figure} [t]
\centering \epsfig{file=./Fig/fig_capacity_different_n_t.eps,
width=0.5\textwidth} \caption{Under total power constraints over
all transmit antennas \eqref{power_cons1}, the secrecy capacities
versus SNRs with different number of antennas $(n_r \geq n_t)$.}
\label{fig_capacity_different_n_t}
\end{figure}
\begin{figure} [t]
\centering \epsfig{file=./Fig/fig_capacity_different_n_e.eps,
width=0.5\textwidth} \caption{Under total power constraints over
all transmit antennas \eqref{power_cons1}, the secrecy capacities
versus SNRs with single transmit antenna $(n_t=1)$.}
\label{fig_capacity_different_n_e}
\end{figure}

\begin{figure} [t]
\centering \epsfig{file=./Fig/fig_alpha.eps, width=0.5\textwidth}
\caption{Under total power constraints over all transmit antennas
\eqref{power_cons1}, the MIMOME secrecy rates $R_s(\alpha
\mathbf{I})$ in \eqref{eq_capacity_object} (with channel input
matrix $\Sigma_{{\mathbf x}}=\alpha {\mathbf I}$) versus $\alpha$ where $0 \leq
\alpha \leq P/n_t$.} \label{fig_alpha}
\end{figure}

In Fig. \ref{fig_MIMOME_vs_MISOSE} and
\ref{fig_linesearch_interation_times}, we studies the wiretap
channels under the per-antenna power constraints
\eqref{eq_per_power_cons}. In Fig.\ref{fig_MIMOME_vs_MISOSE}, the
secrecy rates of wiretap channels with $(n_t,n_r,n_e)=(2,2,1)$
(the objection function of \eqref{eq_per_MIMOME_capacity_old} with
channel input covariance matrices $\Sigma_{{\mathbf x}}=diag(P_1,P_2)$)
are compared with the secrecy capacities $C^{p}_s$ of wiretap
channels with $(n_t,n_r,n_e)=(2,1,1)$ (characterized in Theorem
\ref{Theorem_Per}). As predicted in Corollary
\ref{coro_per_SNR_MISOSE}, the MISOSE channels under constraints
\eqref{eq_per_power_cons} cannot have SNR scaling. However, as
stated in Corollary \ref{coro_per_SNR}, the SNR scaling can be
obtained for the wiretap channel with $(n_t,n_r,n_e)=(2,2,1)$ even
we use the suboptimal $\Sigma_{{\mathbf x}}$ to compute the secrecy rate
in Fig.\ref{fig_MIMOME_vs_MISOSE}. In
Fig.\ref{fig_linesearch_interation_times}, we use the algorithm in
Section \ref{Sec_per_antenna} to numerically solve the secrecy
capacity optimization problem \eqref{eq_per_MIMOME_capacity_old}
subject to constraint \eqref{eq_per_power_cons}, where
$(n_t,n_r,n_e)=(2,2,1)$ and $\epsilon=10^{-4}$. The SNR is
$P_1+P_2$ in dB scale. The proposed algorithms converge in few
iterations. Note that the final converged values are very close
the secrecy rates computed using channel input covariance matrices
$\Sigma_{{\mathbf x}}=(P_1,P_2)$. This fact implies that the secrecy rates
for $(n_t,n_r,n_e)=(2,2,1)$ plotted in Fig.
\ref{fig_MIMOME_vs_MISOSE} are very close to the secrecy
capacities under per-antenna power constraints
\eqref{eq_per_power_cons}.

Finally, in Fig. \ref{fig_total_P_vs_per_antenna_P}, we compare
$C^{\;t}_s$ and $C^p_s$, the MISOSE secrecy capacities under the
total power constraints \eqref{power_cons1} and per-antenna power
constraints \eqref{eq_per_power_cons} in Theorem
\ref{Theorem_No_CQI} and \ref{Theorem_Per}, respectively. Since
under total power constraints \eqref{power_cons1}, the
transmitters are free to allocate power between transmit antennas,
the corresponding secrecy capacities are higher than those under
per-antennas power constraints \eqref{eq_per_power_cons}
($C^{\;t}_s \geq C^p_s$ when $P=P_1+P_2$). We also observe that
when the ratio $P_1/P_2$ are larger, the secrecy capacities
$C^p_s$ become smaller when $(n_t, n_r, n_e)=(2,1,1)$. This is due
to that the multiple transmit antennas act more like a single
antenna when $P_1/P_2$ is larger. Similar observations can be
obtained from the secrecy rates under power constraints
\eqref{eq_per_power_cons} and $(n_t, n_r, n_e)=(2,2,1)$ in Fig.
\ref{fig_MIMOME_vs_MISOSE}.

\begin{figure} [t]
\centering \epsfig{file=./Fig/fig_MIMOME_vs_MISOSE.eps,
width=0.5\textwidth} \caption{Under per-antenna power constraints
\eqref{eq_per_power_cons}, secrecy rates of wiretap channels with
$(n_t,n_r,n_e)=(2,2,1)$ versus secrecy capacities of wiretap
channels with $(n_t,n_r,n_e)=(2,1,1).$}
\label{fig_MIMOME_vs_MISOSE}
\end{figure}

\begin{figure} [t]
\centering \epsfig{file=./Fig/fig_linesearch_interation_times.eps,
width=0.5\textwidth} \caption{Convergence of the proposed
iterative algorithms for computing the secrecy capacities under
constraints \eqref{eq_per_power_cons}, where
$(n_t,n_r,n_e)=(2,2,1)$.} \label{fig_linesearch_interation_times}
\end{figure}

\begin{figure} [t]
\centering \epsfig{file=./Fig/fig_total_P_vs_per_antenna_P.eps,
width=0.5\textwidth} \caption{MISOSE secrecy capacities under the
total power constraints \eqref{power_cons1} and per-antennas power
constraints \eqref{eq_per_power_cons}, where
$(n_t,n_r,n_e)=(2,1,1)$.} \label{fig_total_P_vs_per_antenna_P}
\end{figure}

\section{Conclusion}\label{Sec_conclusion}
In this paper, under two different power constraints, the secrecy
capacities in fast fading Rayleigh MIMOME wiretap channels with
only the statistics of CSIT of both the legitimate and
eavesdropper channels were considered. When antennas of the
legitimate receiver were more than (or equal to) those of the
eavesdropper, under the total power constraint, we fully
characterized the MIMOME secrecy capacity. Under the per-antenna
power constraint, we also showed the secrecy capacity for the
MISOSE channel. These results are the first secrecy capacity
results for multiple antenna wiretap channels with partial CSIT.

redpajama_set_name: RedPajamaArXiv


It's been a relatively quiet few days in terms of Premier League news (understandably so given the focus on international friendlies and World Cup qualifying matches), but here are several stories I picked that you may be interested in reading. Check back here daily at EPL Talk for more stories throughout the day, and have a great week.
English Premier League Coverage Is Ten Times More Popular Than Any Other Content (IPTV News). Hong Kong's Now TV has almost one million subscribers to its IPTV product. And according to an executive at Now TV's owners PCCW, the Premier League games are ten times more popular than any other content. Plus, they show every single game.
McCarthy's No Genius, So How Come Wolves Are Going Up? (Championship Talk). There are still several weeks to go in the season, but it looks likely that Wolverhampton Wanderers will return to the Premier League next season. Writer Dan Trelfer examines the secret to manager Mick McCarthy's success.
2009 Premier League Unsung Heroes XI (Soccerlens). Liverpudlian Neil Jones puts together an all-star Premier League team made up of eleven unsung heroes. The only pick I disagree with is Ashley Young, not because he isn't a hero but because I feel he's a lot more established and is by no means unsung. Everyone has been singing his praises for the past couple of seasons. Take a look at the list and see who you think should have been on it.
KSC Fans Take A Walk On The Wild Side (Bundesliga Talk). Read this interesting account by Joe Ashmore of how German police recently used heavyhanded tactics against Karlsruhe supporters, and how Karlsruhe fans reacted.
El Salvador 2-2 USA: CONCACAF Personified (Major League Soccer Talk). Kartik Krishnaiyer dissects the United States' performance from this past weekend's World Cup qualifying match. The comments left by readers range from disgust at the USA's tactics to appreciation of what a cauldron the stadium in San Salvador was.
England's Movement Bodes Well (The Guardian). Who would have thought that Steven Gerrard's plum position for England was on the left wing? Fabio Capello may have finally found how to combine the wonderful talents of Gerrard, Frank Lampard and Wayne Rooney — with help from Ashley Cole — to create an explosive left-sided attack where England have always been culprit. David Pleat explores.
A Few Highlights From This Weekend's International Matches (World Cup Blog). I don't know about you, but I had good intentions of watching a ton of football this past weekend, but I missed out on many of the matches (other than seeing Wales succumb to defeat against Finland, and England master their game versus Slovakia). Thankfully, Laurie over at the World Cup Blog has picked some of the best highlights you (and I) may have missed from this past weekend.
For more Premier League-related news stories, updated throughout the day each and every day, read or subscribe to EPL Talk's Shared Items.


text:
No History of Indian Popular music of the yesteryears can be complete without mentioning Binaca Geet Mala. Binaca Geet Mala was a hugely popular weekly radio countdown show of top filmi songs from Indian cinema listened to by millions of Hindi music lovers, that was broadcast on Radio Ceylon from 1952 to 1994. It was the first radio countdown show of Indian film songs, and has been quoted as being the most popular radio program in India during its run. It was the most popular radio program before Satellite television took over in India. Binaca Geetmala, and its subsequent incarnations Cibaca Sangeetmala, Cibaca Geetmala, and Colgate Cibaca Sangeetmala on Vividh Bharati ran from 1952 to 1994, with annual year-end lists airing from 1954 to 1993.
BUY RECORD
The show was hosted during its entire run by Ameen Sayani, and was very popular in India, with estimated listenership ranging from 900,000 to 2,000,000. It greatly increased the popularity of Radio Ceylon, making it the primary source of popular film music on radio for the Indian subcontinent. The show later aired on Vividh Bharati after 1988, and used to be on for 1/2 hour on Monday night.
At its onset in 1952, the program did not rank songs, but rather played seven contemporary songs in no particular order. Later, the program started ranking the most popular Hindi film songs. The list was compiled initially on the basis of a combination of sales of records in India and listener votes to Radio Ceylon. The popularity was gauged on the basis of record sales, record store owners verdicts and that "shrota-sanghs"(Hindi for listeners clubs). These "shrota-sanghs" would send in the popular songs every week. The clubs were formed because it was possible that a record could sell out of stores quickly and the store would be out of that records, so it wouldn't show up on record sales, despite the record being very popular. The year-end lists were compiled based on points earned by songs through the year (sometimes between 1966 & 70, there would be no points on the weekly shows broadcast, but the year-end shows would be based on point system).
The first countdown show's topper in 1954 was Talat Mahmood's 'Jayen to jayen kahan' from the film Taxi Driver. Top songs from 1954 to 1960:
1954: Jayen to Jayen Kahan
1955: Mera Joota Hai Japani
1956: Ae Dil Hai Mushkil
1957: Zara Samne to Aao
1958: Hai Apna Dil Jo Awara
1959: Haal Kaisa Hai Janaab ka
1960: Zindagi Bhar Nahin Bhoolegi woh barsaat ki raat
Jazzy Joe with Ameen Sayani
MORE BINACA
GEETMALA CHARTS -

redpajama_set_name: RedPajamaCommonCrawl


Maloneys and St Ann's Junior School – Bread Week Fun!
We have had a lovely week with Year 5 children at St Ann's Junior School in Virginia Waters, learning all about bread, its origins, its production, marketing and selling.
This comes about on an annual basis, as part of our commitment to our local schools programme; we introduce a week-long, national curriculum- based set of resources for teachers to educate their pupils in understanding the life-cycle of bread.
The winning group was: FIVE MINI CHEFS, who then sold their rolls in our Maloney's Budgens Virginia Water shop on Friday morning and made £22.50. This will go towards helping pay for something special at school.


At the beginning, there was Flash.
Videos, banner ads, animations, streams – Flash movies and clips have been used on the internet for a long, long time. While the FLV format may be on the way out, it has still been a video standard for internet users all over the world.


text:
"""CCXT: CryptoCurrency eXchange Trading Library"""

# MIT License
# Copyright (c) 2017 Igor Kroitor
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
# The above copyright notice and this permission notice shall be included in all
# copies or substantial portions of the Software.
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
# SOFTWARE.

# ----------------------------------------------------------------------------

__version__ = '2.2.27'

# ----------------------------------------------------------------------------

from ccxt.base.exchange import Exchange                     # noqa: F401
from ccxt.base.precise import Precise                       # noqa: F401

from ccxt.base.decimal_to_precision import decimal_to_precision  # noqa: F401
from ccxt.base.decimal_to_precision import TRUNCATE              # noqa: F401
from ccxt.base.decimal_to_precision import ROUND                 # noqa: F401
from ccxt.base.decimal_to_precision import ROUND_UP              # noqa: F401
from ccxt.base.decimal_to_precision import ROUND_DOWN            # noqa: F401
from ccxt.base.decimal_to_precision import DECIMAL_PLACES        # noqa: F401
from ccxt.base.decimal_to_precision import SIGNIFICANT_DIGITS    # noqa: F401
from ccxt.base.decimal_to_precision import TICK_SIZE             # noqa: F401
from ccxt.base.decimal_to_precision import NO_PADDING            # noqa: F401
from ccxt.base.decimal_to_precision import PAD_WITH_ZERO         # noqa: F401

from ccxt.base import errors
from ccxt.base.errors import BaseError                                # noqa: F401
from ccxt.base.errors import ExchangeError                            # noqa: F401
from ccxt.base.errors import AuthenticationError                      # noqa: F401
from ccxt.base.errors import PermissionDenied                         # noqa: F401
from ccxt.base.errors import AccountNotEnabled                        # noqa: F401
from ccxt.base.errors import AccountSuspended                         # noqa: F401
from ccxt.base.errors import ArgumentsRequired                        # noqa: F401
from ccxt.base.errors import BadRequest                               # noqa: F401
from ccxt.base.errors import BadSymbol                                # noqa: F401
from ccxt.base.errors import MarginModeAlreadySet                     # noqa: F401
from ccxt.base.errors import BadResponse                              # noqa: F401
from ccxt.base.errors import NullResponse                             # noqa: F401
from ccxt.base.errors import InsufficientFunds                        # noqa: F401
from ccxt.base.errors import InvalidAddress                           # noqa: F401
from ccxt.base.errors import AddressPending                           # noqa: F401
from ccxt.base.errors import InvalidOrder                             # noqa: F401
from ccxt.base.errors import OrderNotFound                            # noqa: F401
from ccxt.base.errors import OrderNotCached                           # noqa: F401
from ccxt.base.errors import CancelPending                            # noqa: F401
from ccxt.base.errors import OrderImmediatelyFillable                 # noqa: F401
from ccxt.base.errors import OrderNotFillable                         # noqa: F401
from ccxt.base.errors import DuplicateOrderId                         # noqa: F401
from ccxt.base.errors import NotSupported                             # noqa: F401
from ccxt.base.errors import NetworkError                             # noqa: F401
from ccxt.base.errors import DDoSProtection                           # noqa: F401
from ccxt.base.errors import RateLimitExceeded                        # noqa: F401
from ccxt.base.errors import ExchangeNotAvailable                     # noqa: F401
from ccxt.base.errors import OnMaintenance                            # noqa: F401
from ccxt.base.errors import InvalidNonce                             # noqa: F401
from ccxt.base.errors import RequestTimeout                           # noqa: F401
from ccxt.base.errors import error_hierarchy                          # noqa: F401

from ccxt.aax import aax                                              # noqa: F401
from ccxt.alpaca import alpaca                                        # noqa: F401
from ccxt.ascendex import ascendex                                    # noqa: F401
from ccxt.bequant import bequant                                      # noqa: F401
from ccxt.bibox import bibox                                          # noqa: F401
from ccxt.bigone import bigone                                        # noqa: F401
from ccxt.binance import binance                                      # noqa: F401
from ccxt.binancecoinm import binancecoinm                            # noqa: F401
from ccxt.binanceus import binanceus                                  # noqa: F401
from ccxt.binanceusdm import binanceusdm                              # noqa: F401
from ccxt.bit2c import bit2c                                          # noqa: F401
from ccxt.bitbank import bitbank                                      # noqa: F401
from ccxt.bitbay import bitbay                                        # noqa: F401
from ccxt.bitbns import bitbns                                        # noqa: F401
from ccxt.bitcoincom import bitcoincom                                # noqa: F401
from ccxt.bitfinex import bitfinex                                    # noqa: F401
from ccxt.bitfinex2 import bitfinex2                                  # noqa: F401
from ccxt.bitflyer import bitflyer                                    # noqa: F401
from ccxt.bitforex import bitforex                                    # noqa: F401
from ccxt.bitget import bitget                                        # noqa: F401
from ccxt.bithumb import bithumb                                      # noqa: F401
from ccxt.bitmart import bitmart                                      # noqa: F401
from ccxt.bitmex import bitmex                                        # noqa: F401
from ccxt.bitopro import bitopro                                      # noqa: F401
from ccxt.bitpanda import bitpanda                                    # noqa: F401
from ccxt.bitrue import bitrue                                        # noqa: F401
from ccxt.bitso import bitso                                          # noqa: F401
from ccxt.bitstamp import bitstamp                                    # noqa: F401
from ccxt.bitstamp1 import bitstamp1                                  # noqa: F401
from ccxt.bittrex import bittrex                                      # noqa: F401
from ccxt.bitvavo import bitvavo                                      # noqa: F401
from ccxt.bkex import bkex                                            # noqa: F401
from ccxt.bl3p import bl3p                                            # noqa: F401
from ccxt.blockchaincom import blockchaincom                          # noqa: F401
from ccxt.btcalpha import btcalpha                                    # noqa: F401
from ccxt.btcbox import btcbox                                        # noqa: F401
from ccxt.btcex import btcex                                          # noqa: F401
from ccxt.btcmarkets import btcmarkets                                # noqa: F401
from ccxt.btctradeua import btctradeua                                # noqa: F401
from ccxt.btcturk import btcturk                                      # noqa: F401
from ccxt.buda import buda                                            # noqa: F401
from ccxt.bybit import bybit                                          # noqa: F401
from ccxt.bytetrade import bytetrade                                  # noqa: F401
from ccxt.cex import cex                                              # noqa: F401
from ccxt.coinbase import coinbase                                    # noqa: F401
from ccxt.coinbaseprime import coinbaseprime                          # noqa: F401
from ccxt.coinbasepro import coinbasepro                              # noqa: F401
from ccxt.coincheck import coincheck                                  # noqa: F401
from ccxt.coinex import coinex                                        # noqa: F401
from ccxt.coinfalcon import coinfalcon                                # noqa: F401
from ccxt.coinmate import coinmate                                    # noqa: F401
from ccxt.coinone import coinone                                      # noqa: F401
from ccxt.coinspot import coinspot                                    # noqa: F401
from ccxt.crex24 import crex24                                        # noqa: F401
from ccxt.cryptocom import cryptocom                                  # noqa: F401
from ccxt.currencycom import currencycom                              # noqa: F401
from ccxt.delta import delta                                          # noqa: F401
from ccxt.deribit import deribit                                      # noqa: F401
from ccxt.digifinex import digifinex                                  # noqa: F401
from ccxt.exmo import exmo                                            # noqa: F401
from ccxt.flowbtc import flowbtc                                      # noqa: F401
from ccxt.fmfwio import fmfwio                                        # noqa: F401
from ccxt.gate import gate                                            # noqa: F401
from ccxt.gateio import gateio                                        # noqa: F401
from ccxt.gemini import gemini                                        # noqa: F401
from ccxt.hitbtc import hitbtc                                        # noqa: F401
from ccxt.hitbtc3 import hitbtc3                                      # noqa: F401
from ccxt.hollaex import hollaex                                      # noqa: F401
from ccxt.huobi import huobi                                          # noqa: F401
from ccxt.huobijp import huobijp                                      # noqa: F401
from ccxt.huobipro import huobipro                                    # noqa: F401
from ccxt.idex import idex                                            # noqa: F401
from ccxt.independentreserve import independentreserve                # noqa: F401
from ccxt.indodax import indodax                                      # noqa: F401
from ccxt.itbit import itbit                                          # noqa: F401
from ccxt.kraken import kraken                                        # noqa: F401
from ccxt.kucoin import kucoin                                        # noqa: F401
from ccxt.kucoinfutures import kucoinfutures                          # noqa: F401
from ccxt.kuna import kuna                                            # noqa: F401
from ccxt.latoken import latoken                                      # noqa: F401
from ccxt.lbank import lbank                                          # noqa: F401
from ccxt.lbank2 import lbank2                                        # noqa: F401
from ccxt.liquid import liquid                                        # noqa: F401
from ccxt.luno import luno                                            # noqa: F401
from ccxt.lykke import lykke                                          # noqa: F401
from ccxt.mercado import mercado                                      # noqa: F401
from ccxt.mexc import mexc                                            # noqa: F401
from ccxt.mexc3 import mexc3                                          # noqa: F401
from ccxt.ndax import ndax                                            # noqa: F401
from ccxt.novadax import novadax                                      # noqa: F401
from ccxt.oceanex import oceanex                                      # noqa: F401
from ccxt.okcoin import okcoin                                        # noqa: F401
from ccxt.okex import okex                                            # noqa: F401
from ccxt.okex5 import okex5                                          # noqa: F401
from ccxt.okx import okx                                              # noqa: F401
from ccxt.paymium import paymium                                      # noqa: F401
from ccxt.phemex import phemex                                        # noqa: F401
from ccxt.poloniex import poloniex                                    # noqa: F401
from ccxt.probit import probit                                        # noqa: F401
from ccxt.qtrade import qtrade                                        # noqa: F401
from ccxt.ripio import ripio                                          # noqa: F401
from ccxt.stex import stex                                            # noqa: F401
from ccxt.therock import therock                                      # noqa: F401
from ccxt.tidex import tidex                                          # noqa: F401
from ccxt.timex import timex                                          # noqa: F401
from ccxt.tokocrypto import tokocrypto                                # noqa: F401
from ccxt.upbit import upbit                                          # noqa: F401
from ccxt.wavesexchange import wavesexchange                          # noqa: F401
from ccxt.wazirx import wazirx                                        # noqa: F401
from ccxt.whitebit import whitebit                                    # noqa: F401
from ccxt.woo import woo                                              # noqa: F401
from ccxt.yobit import yobit                                          # noqa: F401
from ccxt.zaif import zaif                                            # noqa: F401
from ccxt.zb import zb                                                # noqa: F401
from ccxt.zipmex import zipmex                                        # noqa: F401
from ccxt.zonda import zonda                                          # noqa: F401

exchanges = [
    'aax',
    'alpaca',
    'ascendex',
    'bequant',
    'bibox',
    'bigone',
    'binance',
    'binancecoinm',
    'binanceus',
    'binanceusdm',
    'bit2c',
    'bitbank',
    'bitbay',
    'bitbns',
    'bitcoincom',
    'bitfinex',
    'bitfinex2',
    'bitflyer',
    'bitforex',
    'bitget',
    'bithumb',
    'bitmart',
    'bitmex',
    'bitopro',
    'bitpanda',
    'bitrue',
    'bitso',
    'bitstamp',
    'bitstamp1',
    'bittrex',
    'bitvavo',
    'bkex',
    'bl3p',
    'blockchaincom',
    'btcalpha',
    'btcbox',
    'btcex',
    'btcmarkets',
    'btctradeua',
    'btcturk',
    'buda',
    'bybit',
    'bytetrade',
    'cex',
    'coinbase',
    'coinbaseprime',
    'coinbasepro',
    'coincheck',
    'coinex',
    'coinfalcon',
    'coinmate',
    'coinone',
    'coinspot',
    'crex24',
    'cryptocom',
    'currencycom',
    'delta',
    'deribit',
    'digifinex',
    'exmo',
    'flowbtc',
    'fmfwio',
    'gate',
    'gateio',
    'gemini',
    'hitbtc',
    'hitbtc3',
    'hollaex',
    'huobi',
    'huobijp',
    'huobipro',
    'idex',
    'independentreserve',
    'indodax',
    'itbit',
    'kraken',
    'kucoin',
    'kucoinfutures',
    'kuna',
    'latoken',
    'lbank',
    'lbank2',
    'liquid',
    'luno',
    'lykke',
    'mercado',
    'mexc',
    'mexc3',
    'ndax',
    'novadax',
    'oceanex',
    'okcoin',
    'okex',
    'okex5',
    'okx',
    'paymium',
    'phemex',
    'poloniex',
    'probit',
    'qtrade',
    'ripio',
    'stex',
    'therock',
    'tidex',
    'timex',
    'tokocrypto',
    'upbit',
    'wavesexchange',
    'wazirx',
    'whitebit',
    'woo',
    'yobit',
    'zaif',
    'zb',
    'zipmex',
    'zonda',
]

base = [
    'Exchange',
    'Precise',
    'exchanges',
    'decimal_to_precision',
]

__all__ = base + errors.__all__ + exchanges

redpajama_set_name: RedPajamaGithub


text:
News from the Seventh Circle of Hell: Sonic the Hedgehog Live Action Form Revealed
...and it's so much worse than we could have imagined.
Graphic design company Hamagami/Carroll, Inc. apparently leaked the appearance of the live-action rendition of Sega's trademark blue hedgehog in his upcoming movie today, in a post that's since been mysteriously deleted from their site. The company mistakenly released the appearance by publishing what's supposedly a guide on how the character is to look in the upcoming film. This post described it as "a new modern take on the brand without abandoning its core essence" and included several poses of the character, one of which mimics his signature finger wag that graced the title screens for the original games on the Sega Genesis.
The notes above include comments on his "brand personality," which describes him as "irreverent and sarcastic," but also "chill and likable." So, maybe we'll hear Sonic use some light cursing? Sonic and his pals haven't exactly had the cleanest mouths in gaming, so I'm sure it isn't entirely out of the question.
Paramount Pictures is set to launch Sonic the Hedgehog into cinemas this November, starring James Marsden, Ben Schwartz as Sonic, and Jim Carrey as Dr. Eggman of all people. Every generation gets their Super Mario Bros. Movie. Apparently Sonic's time has come.
Not coming anytime soon: Sonic's gloves (which have apparently been replaced with white fur, I guess?) We'll keep up to date as this disasterpiece unfolds, finally culminating (hopefully) with a full review upon its release. Until then, keep running, Sonic. Far, far away from any studio executives...
Tags: 2019, ben scwartz, james marsden, jim carrey, movie sleuth, news, november, paramount pictures, sonic the hedgehog, theater, video game, wes ball at 3/04/2019


When many internationals relocate, they find themselves in a new place, knowing few people and, in the case of spouses or partners of those relocated for work, with some extra time on their hands. It can often be intimidating to step out of your comfort zone in a new place but volunteering is a great way not only for expats to get involved and give a helping hand in their new local community but also an ideal means to immerse themselves in the Dutch culture and meet new people with similar interests.
The Netherlands is the perfect place for internationals looking to get involved with volunteering as there are an abundance of opportunities that require little or no Dutch language skills. Some might be apprehensive to sign up as they are unsure time-wise of the commitment they can dedicate to their chosen cause. Not to worry, many volunteering opportunities are flexible and require no fixed commitment but merely just signing up on the days and times that they can, in the end any help is better than no help as all.
The great thing about volunteering is that there is something for everyone. Whether you are interested in helping educate the underprivileged, lending a listening ear to the elderly while taking them for a stroll in the park, whether you enjoy the environment and animals, sports or food preparation, the major Dutch cities provide numerous and diverse ways to get involved as a volunteer in the local community.
Don't know how to get involved? No worries! Expat Republic has put together a list of the various volunteer opportunities and the city-specific links for Amsterdam, Rotterdam, The Hague, Utrecht and Eindhoven to help you take the first step towards getting involved.
From volunteering in shop fronts to raise money for refugees to jewellery making and other craft activities with the elderly, Amsterdam is full of opportunities to get involved and make a difference. For those more comfortable around food, there is meal preparation for low or no income earners in the local community as well as packing boxes at the Food Bank. There is the possibility to assist with literacy and education through homework classes and also the opportunity to aid in teaching children of all ages various sports and games with the main focus being inclusion and having fun.
Rotterdam offers a diverse range of volunteering activities. Examples include packing birthday boxes to brighten the special day of underprivileged children, helping with the cultivation of fruit and vegetables to provide healthy food for those in need, dancing and walking around the city with the elderly and yoga with the homeless.
None of these activities require a fixed term commitment so you can participate and lend a hand whenever it fits with your schedule. There are also opportunities to assist at the evening school for the mentally disabled and for film fanatics, the International Film Festival of Rotterdam annually seeks volunteers to help out in the box office and backstage.
For animal lovers, The Hague is a good place for animal loving volunteers. Examples include dog training volunteers and also people to help out at an urban farm assisting children as they visit, learn about and touch the animals.
There is also the need for photographers and website builders. Opportunities for volunteers to help with sorting and keeping shelves stocked at the Give Away shop for the underprivileged and those willing to help out with fundraising activities.
Also in The Hague, just as in the other cities, there is a demand for volunteers to keep the elderly company by taking them for walks, serving them food and keeping them company.
Utrecht is the perfect place for sporty volunteers. Why not play table tennis or assist as a football trainer for people with disabilities? Or if you prefer the pool there are opportunities to take women and children swimming. There is a demand for people to organise activities and events at the shelter for refugees. You can also prepare food for the elderly.
If you'd like to assist kids with their homework, then Utrecht has some options. If you prefer being outside, then you can aid with playground maintenance, gardening and painting. As for the cat people among the expat community, there is also the need for volunteer feline foster carers.
For the linguists, the Eindhoven community is looking for people to run various language courses, including English and Spanish. Instead if you have a green thumb, you can offer a helping hand cultivating fruit and vegetables for the Eindhoven Food Bank. Why not share your passion for sport with someone old and lonely and watch a soccer match with the elderly? It's as simple as turning on a T.V., making a cup of coffee and chatting about the game. And if you're already Dutch enough to know your way around a bicycle, help out in teaching refugees how to fix bikes.
So, what are you waiting for? It's as simple as clicking on one of the links provided and signing up. Make a difference in your new neighborhood, get involved and make friends along the way.
Please keep in mind that if you are volunteering, you will need to make sure your residence status allows it.


This article is part of our College Football Draft Kit series.
With the season around the corner, we'll be posting a series of position-by-position writeups this week to get you familiar with some of the top players in the college fantasy football landscape. We're kicking things off with the quarterbacks, where Khalil Tate has the honor of being the No.1 ranked fantasy signal caller in the nation. Behind Tate is an interesting mix of quarterbacks both experienced and new, with some quarterbacks taking over favorable situations and others being tasked with carrying their entire offense on their shoulders.
To see our growing list of projections for over 850 players, including kickers, click here.
From the start of October onward, no quarterback struck fear in opposing defenses like Tate did in 2017. Tate had 26 total touchdowns over the final nine weeks of the season and racked up 1,325 rushing yards in that stretch. Now that Tate enters the year as Arizona's unquestioned leader, he'll rightfully draw plenty of Heisman buzz. New coach Kevin Sumlin's track record in developing dual-threat quarterbacks only bolsters Tate's stock heading into 2018. There's no other quarterback that can take over a game like Tate, which makes him appointment viewing every week.


text:
Jason Bryant, Mat Talk Podcast Network
Mat Talk Podcast Network
Dylan Wanagiel breaks down the first Garden State Grapple on November 27
Dylan Wanagiel, the Vice President of Sports Properties and Special Events for The Prudential Center in Newark, New Jersey comes on the show before he cooks up some turkey to talk about the first Garden State Grapple on November 27. Wanagiel brought wrestling to Madison Square Garden with the Grapple at the Garden and was part of the group that brought the NCAA Division I Championships to MSG back in 2016. He's working to build a new event in New Jersey at his new building and features four Division I schools - Wisconsin, Rutgers, North Carolina and Hofstra - as well as combat sports, youth wrestling and elite high school wrestling across four mats at the home of the NHL's New Jersey Devils. College streaming live on Big Ten Network, show up on Saturday for $25. Happy Thanksgiving! Links to Follow Daily Wrestling Newsletter: https://www.mattalkonline.com/news Contribute: https://www.mattalkonline.com/contribute Patreon: https://www.patreon.com/mattalkonline Rokfin: https://www.rokfin.com/creator/mattalkonline The Short Time Time Wrestling Podcast is proudly supported by Compound Sportswear: https://www.mattalkonline.com/compound Quick Subscribe: https://www.Podfollow.com/shorttime Short Time Wrestling Podcast: Episode 690 - November 25, 2021
Release Date: a month ago
dylan wanagiel wanagiel madison square garden sports properties the ncaa division i championships newark new jersey turkey november 27 first brought wrestling wanagiel new jersey sports school building november event division live garden happy