text:
\section{Introduction}
With the fast development of the high speed train (HST) networks around the world, particularly in China, Japan and Europe, provisioning high speed wireless connections to railway passengers has become an increasingly pressing issue to be tackled. However, moving at a speed of 300-500km/h poses significant difficulties for the design of efficient and reliable wireless systems due to two major obstacles, i.e., extremely frequent handovers and fast variation of the channel conditions \cite{calle2013long}. One of the promising solutions to address these challenges is to exploit the spatial diversity by implementing an antenna array on the top of the train \cite{luo2013efficient,ghazal2012non}.

The potential benefits offered by multiple antenna technology depend heavily on the availability of channel state information (CSI) at the communication nodes. Nevertheless, in the high mobility scenario, obtaining accurate CSI is a difficult task, hence a natural choice is to adopt the differential space-time modulation (DSTM) scheme, which eliminates the requirement for continuous tracking of the CSI \cite{hughes2000differential}. Although DSTM for MIMO systems has been well studied in the literature \cite{jafarkhani2001multiple,hochwald2000unitary,hassibi2002cayley}, it is also understood that the performance of DSTM deteriorates substantially in fast fading channels \cite{schober2002noncoherent,liu2004structures,nguyen2012performance}. Therefore, direct application of DSTM for HST communications is not a good option.

Motivated by this, we propose an improved DSTM scheme, which exploits the \emph{joint spatial and temporal channel correlation inherent in moving antenna array platforms}. At this point, it is worth emphasizing that the correlation effect considered in the current work differs significantly from the conventional concepts of \emph{spatial} or \emph{temporal} correlation. In the MIMO literature, see for instance \cite{kermoal2002stochastic,weichselberger2006stochastic}, the term \emph{spatial correlation} in general refers to \emph{antenna correlation} mainly due to the insufficient separation between the antenna elements. However, we assume that the antennas at both the transmitter and receiver are sufficiently separated, as such, there exists no \emph{antenna correlation} between antenna elements. Also, conventionally, the temporal correlation \cite{li1998robust,choi2001channel,ningsun2014spectral} is used to model the channel variation of the same channel at different time instants. Based on this terminology, coherence time and quasi-stationary channel were subsequently formulated. However, under high mobility conditions, the stationary interval of the channel is extremely small. Therefore, the joint spatial and temporal correlation model proposed in this paper is different from those appeared in prior works, it is used to model the relationship between channel realizations of distinct antenna pairs at different time instants due to the mobility of the antenna array. We will elaborate this point later in Section II. Leveraging this new correlation information, the proposed DSTM scheme is shown to achieve robust communications under high mobility conditions.

The main contributions of this paper are summarized as follows:
\begin{itemize}
    \item We discover a novel spatial and temporal correlation pattern inherent to wireless communications systems with moving antenna array. In addition, the correlation structure is derived, and the impact of key physical parameters on channel correlation is characterized.
    \item Capitalizing on the new correlation information, the optimal decoding criterion for the DSTM scheme is derived. In addition, pairwise error probability (PEP) upper-bounds are obtained, which enables the study of the impact of node mobility and transmission block length on the system performance. Analytical results demonstrates that exploiting channel correlation statistics brings a significant performance gain in high mobility settings.
    \item An adaptive transmission method, where the transmitter adjusts the block length adaptively according to the node velocity, is proposed. It was shown that the proposed adaptive scheme yields improved error performance.
\end{itemize}

The remainder of this paper is organized as follows. In Section II, the joint spatial and temporal channel correlation model is presented. An improved DSTM scheme and its error performance is studied in Section III. In Section IV, an adaptive transmission method is proposed for further performance improvement. Finally, Section V concludes the paper.

Throughout the rest of the paper, matrices and vectors are denoted by bold capital and lowercase letters, respectively. Let $\mathbf{V}\in\mathbb{C}^{M\times N}$ mean that the complex matrix $\mathbf{V}$ is consist of $M$ rows and $N$ columns. $\textbf{I}_{N}$ denotes an $N\times N$ identity matrix; $\textrm{E}\left[\cdot\right]$ is the mathematical expectation operator; $\left ( \cdot  \right )^{H}$ stands for Hermitian transpose; $\left\|\cdot\right\|_{\textrm{F}}$ refers to the Frobenius norm; $\mathrm{Re}\left\{\cdot\right\}$ gives the real part; $\mathrm{Tr}\left\{ \cdot \right\}$ is the matrix trace operation; $\left\lfloor \cdot \right\rfloor$ and $\left\lceil \cdot \right\rceil$ denote, respectively, the floor and ceiling functions; $\mathrm{Round}\left\{ \cdot \right\}$ rounds the element to the nearest integer.

\section{Joint Spatial and Temporal Channel Correlation Model}

We consider the HST communication scenario\footnote{While this paper focuses on the HST scenario, the proposed correlation model also applies to other high mobility scenarios.}, where the base station (BS) equipped with $N_{\textrm{T}}$ antennas sends information to passengers with the help of a dedicated relay equipped with $N_{\textrm{R}}$ antennas with uniform spacing $D$ mounted on top of the train, as illustrated in Fig. \ref{Fig_Scenario}. Direct links between the BS and passengers do not exist due to heavy penetration loss of the radio signal through the well shield carriages \cite{3gpp2014mobile}. As such, access points (AP) are installed inside the train to provide network services for passengers. Since the communication links between the AP and passengers are almost static, establishing high speed connections is relatively simple. Hence, in this paper, we focus on the more challenging point-to-point communication link between the BS and relay.

\begin{figure}[ht]\centering
\includegraphics[angle=0,scale=0.1]{Scenario.eps}
\caption{Schematic model for the HST scenario.}
\label{Fig_Scenario}
\end{figure}

At time $t_{i}$, the BS transmits an information block (denoted as block $i$) containing $M$ codeword matrices of size $N_{\textrm{T}}\times T$, as illustrated in Fig. \ref{Fig_Transmission}. Now, denoting the $j$-th codeword matrix in block $i$ as $\textbf{X}_{i,j}$, then the $j$-th matrix observed at the receiver in the $i$-th block is given by
\begin{equation}
\textbf{Y}_{i,j}=\sqrt{P}\textbf{H}_{i,j}\textbf{X}_{i,j}+\textbf{N}_{i,j},
\label{Received-Symbol}
\end{equation}
where $\textbf{Y}_{i,j}\in \mathbb{C}^{N_{\textrm{R}} \times T}$ is the received codeword matrix, $\textbf{H}_{i,j}\in \mathbb{C}^{N_{\textrm{R}} \times N_{\textrm{T}}}$ is the channel matrix between the BS and Relay. $P$ is the transmit power of each antenna and $\textbf{N}_{i,j}\in \mathbb{C}^{N_{\textrm{R}} \times T}$ is the zero-mean complex Gaussian noise with each column having covariance matrix $\textbf{R}_{\textrm{n}}=\sigma^{2}_{\textrm{n}}\textbf{I}_{N_{\textrm{R}}}$.

In this paper, we assume that the entries of $\textbf{H}_{i,j}$ are independent and identically distributed (i.i.d.) complex Gaussian variables with zero-mean and unit-variance. Please note, such isotropic model is particularly relevant to the scenarios where the propagation environment is dominated by the diffuse counterpart, such as viaduct-1a and cuttings \cite{ai2012radio}. Moreover, we assume that the channel matrix $\textbf{H}_{i,j}$ remains static during one codeword duration $T t_{\textrm{s}}$ with $t_{\textrm{s}}$ being the symbol duration. To this end, it is required that $T t_{\textrm{s}} \leq t_{\textrm{c}}$, where $t_{\textrm{c}}\approx \frac{0.423}{f_{d}}$ is the coherence time and $f_{d}$ represents the maximum Doppler frequency.

\begin{figure}[ht]\centering
\includegraphics[angle=0,scale=0.15]{Transmission.eps}
\caption{Information transmission block.}
\label{Fig_Transmission}
\end{figure}

\subsection{Channel Correlation Model}
We now give a detailed explanation on the channel correlation model adopted in the paper.

\begin{figure}[ht]\centering
\includegraphics[angle=0,scale=0.135]{Mobility_Model.eps}
\caption{Moving antenna array introduces joint spatial and temporal channel correlation.}
\label{Fig_Mobility_Model}
\end{figure}

Consider the moving antenna array depicted in Fig. \ref{Fig_Mobility_Model}. Without loss of generality, we assume that the HST has constant velocity $v \in [0,+\infty)$ and moving direction $\theta \in (-\frac{1}{2}\pi,\frac{1}{2}\pi)$. Intuitively, once $v$ and $\theta$ are specified, the relative position between the antennas at $t_{i+1}$ and $t_{i}$ is determined. Throughout this paper, we assume that $v$ and $\theta$ are perfectly known by the HST. This is reasonable with the aid of mature tracking and positioning techniques such as the Global Positioning System (GPS).

Now let us consider a special case in which the left most antenna at time $t_{i+1}$ arrives at almost the same physical location of the second left most antenna at time $t_i$. Assume that the positions of these two antennas are within half wavelength. Then, combined with the fact that $\tau$ is very short due to high mobility, it can be inferred that the scattering environment remains almost identical. In this sense, it is very likely that the channel between the BS and the first antenna at $t_{i+1}$ is highly correlated with the channel between the BS and the second antenna at $t_i$. With an abuse of terminology, we refer this type of correlation as \emph{spatial-temporal correlation}.\footnote{The observation of the ``spatial-temporal channel correlation'' phenomenon has been independently made in a recent work \cite[Fig. 2]{Nirwan}, where a novel antenna adapting method was proposed to exploit this desirable feature to address the rapid channel variations in HST communication systems.}

Noticing the fact that such spatial-temporal correlation only exists between different codeword blocks, i.e., between channel matrices ${\bf H}_{i,j}$ and ${\bf H}_{i+1,j}$, we drop the subscript $j$ for notational convenience. Now, expressing the channel matrix ${\bf H}_{i}$ into column vector form
\begin{equation}
\textbf{H}_{i}=[\textbf{h}_{1,i}~\textbf{h}_{2,i}~\dots~\textbf{h}_{N_{\textrm{T}},i}],
\label{Channel-Decomposition}
\end{equation}
it is easy to see that the correlation structure between $\textbf{h}_{m,i}$ and $\textbf{h}_{m,i+1}$ is the same as that between $\textbf{h}_{n,i}$ and $\textbf{h}_{n,i+1}$, i.e., $\textrm{E}[\textbf{h}_{m,i+1}\textbf{h}_{m,i}^{H}]=\textrm{E}[\textbf{h}_{n,i+1}\textbf{h}_{n,i}^{H}],~1\leq m,n \leq N_{\textrm{T}}$. Denoting the correlation matrix $\textbf{C}$ as
\begin{equation}
\textbf{C}=\textrm{E}[\textbf{h}_{m,i+1}\textbf{h}_{m,i}^{H}],~1\leq m \leq N_{\textrm{T}},
\label{Correlation-Model}
\end{equation}
then invoking the vector autoregressive (VAR) model \cite{reinsel1992vector,hamilton1994time}, the relation between $\textbf{h}_{m,i+1}$ and $\textbf{h}_{m,i}$ is formulated by
\begin{equation}
\textbf{h}_{m,i+1}=\textbf{C}\textbf{h}_{m,i}+\textbf{u}_{m,i+1},~1\leq m \leq N_{\textrm{T}},
\label{AR1-Model}
\end{equation}
where $\textbf{u}_{m,i+1}\in \mathbb{C}^{N_{\textrm{R}}\times1}$ is a random complex Gaussian noise vector with covariance matrix $\textbf{R}_{\textrm{u}}$. To ensure that $\left\{ \textbf{h}_{m,i} \right\}_{i}$ is a stationary VAR(1) time series, we need $\textbf{R}_{\textrm{u}}=\textbf{I}_{N_{\textrm{R}}}-\textbf{C}\textbf{C}^{H}$.

Combining (\ref{Channel-Decomposition}), (\ref{Correlation-Model}) and (\ref{AR1-Model}), the correlation between channel samples can be rewritten in a compact matrix form as
\begin{equation}
\textbf{H}_{i+1}=\textbf{C}\textbf{H}_{i}+\textbf{U}_{i+1},
\label{Overall-AR1-Model}
\end{equation}
where $\textbf{U}_{i+1}\triangleq[\textbf{u}_{1,i+1}~\textbf{u}_{2,i+1}~\dots~\textbf{u}_{N_{\textrm{T}},i+1}]$.

\newcounter{TempEqCnt1}
\setcounter{TempEqCnt1}{\value{equation}}
\setcounter{equation}{14}
\begin{figure*}[ht]

\begin{equation}
\rho \left ( \tau ,D \right ) = \frac{I_{0}\left( \sqrt{ \kappa^2-a^{2}-b^{2}+2ab\cos\left(\beta - \alpha \right )-j2\kappa \left[ a\cos\left( \mu - \alpha \right)-b\cos\left( \mu-\beta \right ) \right ] } \right)}{I_{0}\left( \kappa \right)}e^{-c_{0}v\left | \tau \right |},
\label{modi_correlation_model}
\end{equation}
\hrulefill
\end{figure*}
\setcounter{equation}{\value{TempEqCnt1}}

\subsection{Structure of the Correlation Matrix ${\bf C}$}
We now study the structure of the correlation matrix $\textbf{C}$. Before going into the details, we first state the following important fact which will be invoked in the ensuing derivations.
\begin{fact}
If the distance between the two antennas are sufficiently large, then there is no spatial-temporal correlation between them.
\end{fact}
This Fact is rather intuitive, and has been adopted in many channel models. For instance, according to the Clark's model, antennas separated by a distance over half wavelength can be regarded as spatially uncorrelated. In the current work, we generalize this Fact to account for the correlation effect between antenna pairs at different time. For instance, the distance between the first left most antenna at time $t_{i}$ and the second left most antenna at time $t_{i+1}$ is larger than half wavelength, hence, there is no spatial-temporal correlation.

Keeping Fact 1 in mind, we are ready to present the following key results:
\begin{lemma}\label{lemma1}
The $N_{\emph{R}} \times N_{\emph{R}}$ correlation matrix ${\bf C}$ can be modeled by a Toeplitz matrix as follows
\begin{equation}
\emph{\textbf{C}}=
\begin{bmatrix}
 \cdots &\rho_{l}  &\rho_{l+1}   &        &\emph{\large{\textbf{0}}}        \\
        &        &\rho_{l}  &\ddots  &        \\
        &        &        &\ddots  &\rho_{l+1}   \\
        &\emph{\large{\textbf{0}}}      &        &        &\rho_{l}  \\
        &        &        &        & \vdots
\end{bmatrix}
,
\label{Toeplitz-correlation-Model}
\end{equation}
where $l$ is some context-specific integer, while $\rho_{l}$ and $\rho_{l+1}$ denote the correlation coefficients, and they are determined by system parameters including block duration $\tau$, velocity $v$, antenna spacing $D$, and moving direction $\theta$, etc.
\end{lemma}
\begin{IEEEproof}[Proof]
See Appendix \ref{proof_lemma1}.
\end{IEEEproof}

\emph{Remarks}: Lemma \ref{lemma1} indicates that at time $t_{i+1}$, the channel gain $h_{m,n,i+1}$ is at most correlated with two entries of $\textbf{h}_{m,i}$, with correlation coefficients $\rho_{l}$ and $\rho_{l+1}$, respectively. This observation is rather intuitive, and is a direct consequence of Fact 1.

Without loss of generality, assume that the distance threshold in Fact 1 is $0.5\lambda$. Then, based on some simple geometry analysis of the spatial-temporal model depicted in Fig. \ref{Fig_Mobility_Model}, the matrix $\textbf{C}$ may exhibit three different structure and properties:

\emph{Case I}: There exists an $l$ that satisfies the following conditions
\begin{equation}
\sqrt{\left( v\tau \cos(\theta)-lD \right)^{2}+\left(v\tau \sin(\theta)\right)^{2}}\leq 0.5\lambda, \mbox{ and}
\label{Correlation_case_a_condition1}
\end{equation}
\begin{equation}
\sqrt{\left( lD+D-v\tau \cos(\theta) \right)^{2}+\left(v\tau \sin(\theta)\right)^{2}}\leq 0.5\lambda.
\label{Correlation_case_a_condition2}
\end{equation}
In such scenarios, the position of the first antenna at $t_{i+1}$, is within half wavelength of the $(l+1)$-th and the $(l+2)$-th antenna at $t_{i}$, implying strong correlation between the corresponding channels as illustrated in Fig. \ref{cases}(a). Based on Fact 1, it can be inferred that
\begin{equation}
\rho_{l}\neq 0,~\rho_{l+1}\neq 0, \mbox{ and }l=\lfloor \frac{v\tau \cos( \theta)}{D} \rfloor.
\label{Correlation_case_a}
\end{equation}

\emph{Case II}: There exists an $l$ that satisfies the following conditions
\begin{equation}
\sqrt{\left( v\tau \cos(\theta)-lD \right)^{2}+\left(v\tau \sin(\theta)\right)^{2}}\leq 0.5\lambda, \mbox{ and}
\label{Correlation_case_b_condition1}
\end{equation}
\begin{equation}
\sqrt{\left( v\tau \cos(\theta)-lD+D \right)^{2}+\left(v\tau \sin(\theta)\right)^{2}}> 0.5\lambda, \mbox{ and}
\label{Correlation_case_b_condition2}
\end{equation}
\begin{equation}
\sqrt{\left( lD+D-v\tau \cos(\theta) \right)^{2}+\left(v\tau \sin(\theta)\right)^{2}}> 0.5\lambda.
\label{Correlation_case_b_condition3}
\end{equation}
In such scenarios, the channel of the first antenna at $t_{i+1}$ is only correlated to the $(l+1)$-th antenna at $t_{i}$ as illustrated in Fig. \ref{cases}(b). Hence, we have
\begin{equation}
\rho_{l}\neq 0, \mbox{ and }l=\mathrm{Round}\left\{ \frac{v\tau \cos( \theta)}{D} \right\}.
\label{Correlation_case_b}
\end{equation}

\emph{Case III}: There exists no such $l$ that satisfies
\begin{equation}
\sqrt{\left( v\tau \cos(\theta)-lD \right)^{2}+\left(v\tau \sin(\theta)\right)^{2}}\leq 0.5\lambda,
\label{Correlation_case_c_condition}
\end{equation}
which indicates that the channel at $t_{i+1}$ is almost uncorrelated to that at $t_{i}$, as illustrated in Fig. \ref{cases}(c) and Fig. \ref{cases}(d). Equivalently, in such scenarios $\textbf{C}$ is approximately all-zero.

\begin{figure}[h!t]
\centering
\subfigure[Case I: $\theta=0$, $v\tau=0.5D$]{
\label{caseI}
\includegraphics[angle=0,scale=0.22]{caseI.eps}}
\hspace{0.15in}
\subfigure[Case II: $\theta=0$, $v\tau=D$]{
\label{caseII}
\includegraphics[angle=0,scale=0.22]{caseII.eps}}
\hspace{0.15in}
\subfigure[Case III: $\theta=0.5\pi$, $v\tau>0.5\lambda$]{
\label{caseIIIa}
\includegraphics[angle=0,scale=0.22]{caseIIIa.eps}}
\hspace{0.15in}
\subfigure[Case III: $\theta=0$, $v\tau=4D$]{
\label{caseIIIb}
\includegraphics[angle=0,scale=0.22]{caseIIIb.eps}}
\caption{Specific examples for the given three cases.}
\label{cases}
\end{figure}

\subsection{A Specific Model for Correlation Coefficients}

In general, the correlation coefficients $\rho_{l}$ and $\rho_{l+1}$ can be estimated from measurement data. However, we may exploit a specific spatial-temporal correlation structure to establish an analytical model. Building upon the channel correlation framework proposed in \cite{abdi2002space}, the moving scattering objects are modeled as poisson point processes (PPP), which results in an exponential decay factor attached to the cross-correlation function. Hence, the final correlation coefficient expression can be obtained as (\ref{modi_correlation_model}), where $I_{0}( z )$ is $0$-th order modified Bessel function of the first kind. $\alpha$, $\beta$ are defined in Fig. \ref{Fig_Mobility_Model}, and $\theta = \beta - \alpha$. Moreover, $a\triangleq 2\pi f_{D}\tau$ and $b\triangleq 2\pi D/\lambda$ where $f_{D}=\frac{v}{\lambda}$ is the maximum Doppler shift and $\lambda$ is the carrier wavelength. The parameter $\kappa$ indicates the width of angle of arrival (AOA) and $\mu\in\left[-\pi, \pi\right)$ accounts for the mean direction of AOA. $c_{0}$ is a real constant characterizing the spatial property of the scattering objects. For the special case of isotropic scattering, we have $\kappa=0$ and $\tau >0$, as such, $\rho_{l}$ and $\rho_{l+1}$ can be expressed as
\setcounter{equation}{15}
\begin{equation}
\rho_{k}=
\left\{\begin{matrix}
J_{0}\left( \sqrt{a^{2}+b_{k}^{2}-2ab_{k}\cos\left( \theta \right)} \right)e^{-c_{0}v \tau },~k=l,l+1
\\
0,~~~~~~~\mbox{otherwise}~~~~~~~~~~~~
\end{matrix}\right.
\label{alpha_and_beta}
\end{equation}
where $J_{0}\left( z \right)$ is $0$-th order Bessel function of the first kind and $b_{k}\triangleq 2\pi kD/\lambda$. Note that a similar channel correlation model has been adopted in \cite{jakes1994microwave} and \cite{truong2013effects}.

As mentioned above, $v$ and $\theta$ are known as priori information at the HST. Then $\rho_{l}$ and $\rho_{l+1}$ can be directly derived from (\ref{alpha_and_beta}). Invoking Lemma $1$, this indicates that $\textbf{C}$ can be obtained by the relay.

\section{A Novel Differential Modulation Scheme}
In this section, we first propose a novel differential modulation scheme to exploit the spatial-temporal correlation structure discovered in the previous section. Then, the error performance of the proposed scheme is studied by deriving an uniform PEP upper bound. Finally, we look at the special case when the correlation matrix $\textbf{C}$ has only one superdiagonal or main diagonal.

\newcounter{TempEqCnt2}
\setcounter{TempEqCnt2}{\value{equation}}
\setcounter{equation}{24}
\begin{figure*}[ht]
\begin{equation}
\textrm{Pr}\left \{ \textbf{G}\rightarrow {\textbf{G}}' \right \}\leq
\frac{1}{\left | \textbf{I}_{N_\textrm{T}}+\frac{\gamma^{2}}{1+2\gamma }\left [ \textbf{I}_{N_\textrm{T}}-\frac{1}{ 4N_{\textrm{T}}}\textbf{D}\left ( \textbf{I}_{N_{\textrm{T}}}+{\textbf{G}}'\textbf{G}^{H} \right )\left ( \textbf{I}_{N_{\textrm{T}}}+\textbf{G}{\textbf{G}}'^{H} \right )\textbf{D}^{H} \right ] \right |^{N_{\textrm{R}}-l}},
\label{PEP}
\end{equation}
\hrulefill
\end{figure*}
\setcounter{equation}{\value{TempEqCnt2}}

\subsection{Proposed Differential Modulation Scheme}
Space-time differential modulation schemes based on orthogonal designs have been extensively studied in the MIMO literature, see for instance \cite{hochwald2000unitary,hughes2000differential,hassibi2002cayley}, where the codeword matrix $\textbf{X}_{i}$ satisfies
the following orthogonality properties:
\begin{equation}
\textbf{X}_{i}\textbf{X}_{i}^H=T\textbf{I}_{N_{\textrm{T}}}, \mbox{ and } \textbf{X}_{i}^H\textbf{X}_{i}=N_{\textrm{T}}\textbf{I}_{T}.
\label{Orthogonal-Design}
\end{equation}

However, the conventional differential modulation schemes are designed for static or quasi-static channels, therefore can not handle the fast fading channel scenarios under high mobility conditions. To tackle this critical issue, we propose an improved differential space-time modulation scheme, exploiting the spatial-temporal correlation structure of the moving antenna array.

For differential transmission, the unitary codeword matrices $\textbf{X}_{0},\textbf{X}_{1},\dots,\textbf{X}_{i},\dots$ are organized in a way similar to DPSK:
\begin{equation}
\begin{aligned}
&\textbf{X}_{0}=\textbf{D},\\
&\textbf{X}_{i+1}=\textbf{X}_{i}\textbf{G}_{k},~\textbf{G}_{k}\in\mathcal{G},~k=1,\dots,K
\end{aligned}\label{Differential-encoding}
\end{equation}
where $\textbf{D}$ is the initial matrix known \emph{a priori} at the receiver, and satisfies $\textbf{D}\textbf{D}^{H}=\textbf{D}^{H}\textbf{D}=N_{\textrm{T}}\textbf{I}_{N_{\textrm{T}}}$ according to (\ref{Orthogonal-Design}), and $\textbf{G}_{k}$ is the information-bearing matrix which belongs to an $N_{\textrm{T}}\times N_{\textrm{T}}$ unitary matrices set $\mathcal{G}$ with cardinality $K$ and is carefully designed to ensure that $\textbf{X}_{i+1}\in \textbf{D}\mathcal{G}$ whenever $\textbf{X}_{i}\in \textbf{D}\mathcal{G}$. Accordingly, $\textbf{D}\mathcal{G}=\left\{ \textbf{D}\textbf{G}:\textbf{G}\in \mathcal{G} \right\}$ is referred to as the \emph{differential code set}.

We now look at the differential detection phase. Substituting (\ref{Overall-AR1-Model}) and (\ref{Differential-encoding}) into (\ref{Received-Symbol}), the received symbol matrices at time $t_i$ and $t_{i+1}$ can be expressed as
\begin{equation}
\begin{split}
\textbf{Y}_{i}~=~&\sqrt{P}\textbf{H}_{i}\textbf{X}_{i}+\textbf{N}_{i}, \mbox{ and }\\
\textbf{Y}_{i+1}~=~&\sqrt{P}\textbf{H}_{i+1}\textbf{X}_{i+1}+\textbf{N}_{i+1}\\
                ~=~&\sqrt{P}\textbf{C}\textbf{H}_{i}\textbf{X}_{i+1}+\sqrt{P}\textbf{U}_{i+1}\textbf{X}_{i+1}+\textbf{N}_{i+1}\\
                ~=~&\sqrt{P}\textbf{C}\textbf{H}_{i}\textbf{X}_{i}\textbf{G}_{k}+\sqrt{P}\textbf{U}_{i+1}\textbf{X}_{i}\textbf{G}_{k}+\textbf{N}_{i+1}.
\label{Two-Received}
\end{split}
\end{equation}

Invoking the analysis in \cite{hughes2000differential}, we have the following theorem.

\begin{theorem}\label{theorem1}
The optimal differential decoding criterion is given by
\begin{equation}
\widehat{\textbf{\emph{G}}}=\mathrm{arg}\underset{\textbf{\emph{G}}\in \mathcal{G}}{\mathrm{max}}\mathrm{Re}\left \{ \mathrm{Tr} \left \{\textbf{\emph{G}}\textbf{\emph{Y}}_{i+1}^{H}\textbf{\emph{C}}\textbf{\emph{Y}}_{i} \right \}\right \}.
\label{Improved-Decoding}
\end{equation}
\end{theorem}
\begin{IEEEproof}[Proof]
See Appendix \ref{proof_theorem1}.
\end{IEEEproof}

For better understanding, we give (\ref{Improved-Decoding}) an \emph{estimator-correlator} interpretation as for the conventional maximum likelihood (ML) differential decoder defined by \cite{hughes2000differential}
\begin{equation}
\widehat{\textbf{G}}~=~\mathrm{arg}\underset{\textbf{G}\in \mathcal{G}}{\mathrm{max}}\mathrm{Re}\left\{ \mathrm{Tr} \left \{\textbf{G}\textbf{Y}_{i+1}^{H}\textbf{Y}_{i} \right \}\right\}.
\label{Standard-Decoding}
\end{equation}
It can be observed that (\ref{Standard-Decoding}) is just a special case of (\ref{Improved-Decoding}) when ${\bf C} = {\bf I}$ and $\textbf{U}_{i} = \bf 0$. For this case, since $\textbf{H}_{i+1}= \textbf{H}_{i}$, $\textbf{Y}_{i+1}$ and $\textbf{Y}_{i}$ can be used to estimate $\sqrt{P}\textbf{H}_i\textbf{X}_{i+1}$ (=$\sqrt{P}\textbf{H}_{i+1}\textbf{X}_{i+1}$) and $\sqrt{P}\textbf{H}_{i}\textbf{X}_{i}$, respectively. Hence, the decoding of $\textbf{G}$ can be completed through maximizing the correlation between $\textbf{Y}_{i+1}$ and $\textbf{Y}_{i}\textbf{G}$.

However, in our studied scenario, since now $\textbf{H}_{i+1} \neq \textbf{H}_{i}$, albeit $\sqrt{P}\textbf{H}_{i}\textbf{X}_{i}$ can still be directly estimated by $\textbf{Y}_{i}$, we need to find an alternate estimate for $\sqrt{P}\textbf{H}_{i}\textbf{X}_{i+1}$ instead of $\textbf{Y}_{i+1}$ in order to decode $\textbf{G}$. According to (\ref{Two-Received}), a column of $\textbf{Y}_{i+1}$ can be expressed as
\begin{equation}
\textbf{y}_{i+1}=\sqrt{P}\textbf{C}\textbf{H}_{i}\textbf{x}_{i+1}+\sqrt{P}\textbf{U}_{i}\textbf{x}_{i+1}+\textbf{n}_{i+1},
\label{Receive-subblock}
\end{equation}
where $\textbf{x}_{i+1}$ and $\textbf{n}_{i+1}$ are the corresponding columns of $\textbf{X}_{i+1}$ and $\textbf{N}_{i+1}$, respectively. Recall that $\textrm{E}[\textbf{H}_{i}\textbf{H}_{i}^{H}]=N_{\textrm{T}}\textbf{I}_{N_{\textrm{R}}}$, $\textrm{E}[\textbf{U}_{i}\textbf{U}_{i}^{H}]=N_{\textrm{T}}(\textbf{I}_{N_{\textrm{R}}}-\textbf{C}\textbf{C}^{H} )$ and $\textrm{E}[\textbf{x}_{i+1}\textbf{x}_{i+1}^{H}]=\textbf{I}_{N_{T}}$, $\textbf{H}_{i}\textbf{x}_{i+1}$ can then be estimated by invoking the minimum mean square error (MMSE) estimator
\begin{align}
&\widehat{\textbf{H}_{i}\textbf{x}_{i+1}}\notag\\
&=\frac{N_{\textrm{T}}}{\sqrt{P}}\textbf{C}^{H}\left( N_{\textrm{T}}\left( \textbf{C}\textbf{C}^{H} + \textbf{I}_{N_{\textrm{R}}}- \textbf{C}\textbf{C}^{H} \right) +\frac{\sigma^{2}_{\textrm{n}}}{P}\textbf{I}_{N_{\textrm{R}}} \right)^{-1}\textbf{y}_{i+1}\notag\\
&=\frac{\sqrt{P} N_{\textrm{T}}}{ P N_{\textrm{T}}+\sigma^{2}_{\textrm{n}} } \textbf{C}^{H}\textbf{y}_{i+1}.
\label{Transmit-subblock-Estimation}
\end{align}
Therefore, the optimal estimate of $\textbf{H}_{i}\textbf{X}_{i+1}$ can be written as
\begin{equation}
\widehat{\textbf{H}_{i}\textbf{X}_{i+1}}=\frac{\sqrt{P} N_{\textrm{T}}}{ P N_{\textrm{T}}+\sigma^{2}_{\textrm{n}} }\textbf{C}^{H}\textbf{Y}_{i+1}.
\label{Block-Estimation}
\end{equation}
Thus replacing $\textbf{Y}_{i+1}$ in (\ref{Standard-Decoding}) by $\textbf{C}^{H}\textbf{Y}_{i+1}$ results in the proposed decoder (\ref{Improved-Decoding}).

\subsection{Uniform PEP Upper-Bound}

We now study the error performance of the proposed differential decoding scheme, and obtain an upper-bound for the PEP. By extending the Chernoff bound of the PEP of the ML receiver for the conventional unitary space-time coding system in a quasi-static channel (cf.\cite{hughes2000differential}, eq.(9)) to our studied channel with the specific spatial-temporal correlation, we have the following result:

\begin{theorem}\label{theorem2}
An upper bound of the PEP of the ML receiver for the proposed differential decoding scheme is given by (\ref{PEP}), where $\gamma$ is the equivalent signal-to-interference-plus-noise ratio (SINR) defined as
\setcounter{equation}{25}
\begin{equation}
\gamma=\frac{ \left ( 1-\frac{l}{N_{\textrm{R}}} \right )\left | \rho_{l} \right |^{2}+\left ( 1-\frac{l+1}{N_{\textrm{R}}} \right )\left | \rho_{l+1} \right |^{2} }
{ 1- \left ( 1-\frac{l}{N_{\textrm{R}}} \right )\left | \rho_{l} \right |^{2}-\left ( 1-\frac{l+1}{N_{\textrm{R}}} \right )\left | \rho_{l+1} \right |^{2} +\frac{\sigma^{2}_{\textrm{n}}}{N_{\textrm{T}}P} }.
\label{SINR}
\end{equation}
\end{theorem}
\begin{IEEEproof}[Proof]
See Appendix \ref{proof_theorem2}.
\end{IEEEproof}

From Theorem \ref{theorem2}, we can get the following important observations:

\emph{a)} Increasing the number of antennas at both the BS and the mobile receiver generally improves the error performance. This is a direct consequence of the facts that increasing $N_{\textrm{T}}$ and $N_{\textrm{R}}$ improves the SINR (as shown in (\ref{SINR})) and the PEP is a monotonically decreasing function with respect to $\gamma$.

\emph{b)} According to (\ref{Correlation_case_b}), if the train tends to slow down, i.e., $v\rightarrow0$, then $l\rightarrow0$. Hence, from (\ref{alpha_and_beta}) and (\ref{SINR}), there is $\rho_{0}\rightarrow1$, $\rho_{1}\rightarrow0$ and $\gamma\rightarrow\frac{N_{\textrm{T}}P}{\sigma^{2}_{\textrm{n}}}$. In such a case, (\ref{PEP}) reduces to the original PEP bound for differential unitary space-time codes (see (\ref{Original-PEP}) in Appendix \ref{proof_theorem2}), which indicates that the proposed scheme attains the same error performance as the conventional scheme.

\emph{c)} In the large $N_{\textrm{R}}$ and small $l$ regime, i.e., $\frac{l}{N_{\textrm{R}}}\rightarrow 0$, (\ref{SINR}) simplifies to $\gamma=\frac{ \left | \rho_{l} \right |^{2}+\left | \rho_{l+1} \right |^{2} }
{ 1- \left( \left| \rho_{l} \right|^{2}+\left | \rho_{l+1} \right|^{2} \right) +\frac{\sigma^{2}_{\textrm{n}}}{N_{\textrm{T}}} }$, which suggests that the PEP is a decreasing function with respect to $\left | \rho_{l} \right |^{2}+\left | \rho_{l+1} \right |^{2}$. Since in effect $\left | \rho_{l} \right |^{2}+\left | \rho_{l+1} \right |^{2}$ well reflects the degree of correlation between successive channel realizations, it indicates that the proposed scheme achieves superior error performance when the channel correlation is high.

To validate the aforementioned key observations, we now perform Monte Carlo simulations to illustrate the SER performance of the proposed scheme. For differential unitary space-time modulation, we take the optimal $4\times 4$ cyclic unitary group code $\mathcal{G}=\left \{ \textbf{I}_{4},\textbf{G},\textbf{G}^{2},\textbf{G}^{3} \right \}$ with a Hadamard initial matrix $\textbf{D}$ (Interested readers are referred to \cite{hughes2003optimal} for details regarding the optimal differential unitary space-time modulations). Hence, here $N_{\textrm{T}}=T=4$. Simulation parameters are summarized in Table I.

\begin{table}[htbp]
\begin{minipage}{0.45\textwidth}
\begin{center}
\caption[]{Simulation parameters}
\begin{tabular}{|c|c|}
\hline
Number of BS antennas $N_{\textrm{T}}$&$4$\\ \hline
Number of HST antennas $N_{\textrm{R}}$&$4$\\ \hline
Length of a codeword matrix $T$&$4$\\ \hline
SNR per receive antenna $=\frac{N_{\textrm{T}}P}{\sigma^{2}_{\textrm{n}}}$&$5$dB\\ \hline
Symbol duration $t_{\textrm{s}}$&$50\upmu$s\\ \hline
Spatial parameter $c_{0}$&$0.1$\\ \hline
Carrier frequency $f_{c}$&$3$GHz\\ \hline
Antenna spacing $D$&$0.5\lambda=5\times 10^{-2}$m\\ \hline
Number of codeword matrices in a block $M$&$1$\\ \hline
Block duration $\tau=M\times Tt_{\textrm{s}}$&$200\upmu$s\\ \hline
Train's velocity $v$&$0$ $\sim$ $200$m/s~\footnote{According to the system settings, coherence time is given by $t_{\textrm{c}}\approx \frac{0.423}{f_{D}}=\frac{4.23\times 10^{-2}}{v}$s. Recall the assumption that $Tt_{\textrm{s}}<t_{\textrm{c}}$, the train velocity must satisfy $v<211.5$m/s. Therefore, the range of $v$ is set to $0$ $\sim$ $200$m/s.}\\ \hline
Train's moving direction $\theta$&$0,~\frac{1}{16}\pi,~\frac{1}{8}\pi,~\frac{1}{4}\pi$\\ \hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

Fig. \ref{Fig_SERvsVelocity1} illustrates the SER of the proposed scheme with different velocities and moving directions. The conventional receiver (\ref{Standard-Decoding}) is employed as a benchmark. It is observed that, at the low mobility scenario (i.e., $v<75 \mbox{m/s}$), the conventional receiver achieves almost the same error performance as the proposed scheme. This is intuitive, since in low $v$ regime, the correlation matrix $\textbf{C}$ can be approximated as $\rho_{l}\textbf{I}_{N_{\textrm{R}}}$, indicating that the improved decoder is equivalent to the conventional one. Nevertheless, as the velocity increases, the error performance of the conventional receiver deteriorates substantially. In contrast, if the velocity is sufficiently large (i.e., $v>135 \mbox{m/s}$), the error performance of the proposed scheme actually improves as the velocity increases. This phenomenon will be elaborated in Section III.C. In addition, we see that the proposed scheme heavily depends on the moving direction. When $\theta > \frac{1}{4}\pi$, the performance gain becomes negligible. The reason is that the relative position has changed significantly, and the channels become almost uncorrelated, hence the proposed scheme only yields negligible performance gain.
\begin{figure}[!t]\centering
\includegraphics[angle=0,scale=0.4]{SER_velocity_angle.eps}
\caption{Impacts of velocity and moving direction on the SER.}
\label{Fig_SERvsVelocity1}
\end{figure}

\subsection{A Special Case}
In practice, the HST moves along the track with relatively constant velocity and the antenna array is always aligned with the direction of movement. Hence, in the ensuing analysis, we focus on the practical setting with a constant $v$ and $\theta=0$. In this sense, the channel correlation parameter given in (\ref{alpha_and_beta}) can be computed by
In practice, the HST moves along the track with relatively constant velocity and the antenna array is always aligned with the direction of movement. Hence, in the ensuing analysis, we focus on the practical setting with a constant $v$ and $\theta=0$. In this sense, the channel correlation parameter given in (\ref{alpha_and_beta}) can be computed by
\begin{equation}
\begin{split}
\rho_{l}&=J_{0}\left( \frac{2\pi}{\lambda}\left( v\tau - lD \right) \right)e^{-c_{0}v \tau }, \mbox{ and }\\
\rho_{l+1}=&
\left\{\begin{matrix}
J_{0}\left( \frac{2\pi}{\lambda}\left( v\tau - (l+1)D \right) \right)e^{-c_{0}v \tau },~\textrm{for case I}
\\
0,~~~~~~~~\textrm{for case II}~~~~~
\end{matrix}\right.
\label{alpha_and_beta2}
\end{split}
\end{equation}
When $ v\tau \approx {lD}$, i.e., the corresponding antenna pairs almost overlap at the same physical position at different time instant as illustrated in Fig. \ref{Fig-matched-velocity}, the correlation matrix ${\bf C} $ exhibits a special structure with only a single superdiagonal entry (i.e., case II). For such system, we have the following important result.

\begin{figure}[h!t]
\centering
\subfigure[$\rho_{0},~\left| v\tau \right| < D-0.5\lambda$]{
\label{Fig-matched-velocity-low}
\includegraphics[angle=0,scale=0.08]{low_v2.eps}}
\hspace{0.24in}
\subfigure[$\rho_{1},~\left| v\tau-D \right| < D-0.5\lambda$]{
\label{Fig-matched-velocity-med}
\includegraphics[angle=0,scale=0.08]{medium_v2.eps}}
\hspace{0.12in}
\subfigure[$\rho_{2},~\left| v\tau-2D \right| < D-0.5\lambda$]{
\label{Fig-matched-velocity-high}
\includegraphics[angle=0,scale=0.08]{high_v2.eps}}
\hspace{0.12in}
\subfigure[$\rho_{3},~\left| v\tau-3D \right| < D-0.5\lambda$]{
\label{Fig-matched-velocity-uhigh}
\includegraphics[angle=0,scale=0.08]{ultra-high_v2.eps}}
\caption{Impact of $v\tau$ on the spatial-temporal correlation structure.}
\label{Fig-matched-velocity}
\end{figure}

\begin{theorem}\label{theorem3}
For the special case where $\textbf{C}$ has one single superdiagonal (or main diagonal), the proposed decoding criterion (\ref{Improved-Decoding}) achieves the same error performance as the conventional differential decoder (\ref{Standard-Decoding}) with only $N_{\textrm{R}}-l$ receiving antennas.
\end{theorem}

\begin{IEEEproof}[Proof]
See Appendix \ref{proof_theorem3}.
\end{IEEEproof}

\emph{Remarks}: Theorem \ref{theorem3} indicates that the penalty of the mobility is the requirement of extra receiving antennas at the relay. As illustrated in Fig. \ref{Fig_Theorem_2} where $M=5$, when $v\approx 0,~50,~100$ and $150$m/s, the proposed decoding criterion achieves the same error performance as an equivalent conventional receiver with only $N_{\textrm{R}}-l$ antennas.

Furthermore, a tighter PEP upper bound can be obtained as follows.
\begin{theorem}
When the correlation matrix ${\bf C}$ has one single superdiagonal (or main diagonal) with entries $\rho_{l}$, the PEP of the proposed differential scheme is upper bounded by
\begin{equation}
\begin{split}
&\textrm{Pr}\left \{ \textbf{G}\rightarrow {\textbf{G}}' \right \}\leq\\
&\frac{1}{\left | \textbf{I}_{N_\emph{T}}+\frac{\left( \frac{2\gamma_{1}\gamma_{2}}{\gamma_{1}+\gamma_{2}} \right)^{2}}{4N_{\emph{T}}\left(1+2\left( \frac{2\gamma_{1}\gamma_{2}}{\gamma_{1}+\gamma_{2}} \right) \right ) } \textbf{D}\left ( \textbf{G}-{\textbf{G}}' \right )\left ( \textbf{G}-{\textbf{G}}' \right )^{H}\textbf{D}^{H} \right |^{N_{\emph{R}}-l}},
\label{PEP-pro}
\end{split}
\end{equation}
where $\gamma_{1}$ and $\gamma_{2}$ are expressed as
\begin{equation}
\gamma_{1}=\frac{N_{\emph{T}}P}{\sigma^{2}_{\emph{n}}},~~~~~
\gamma_{2}=\frac{\left| \rho_{l} \right|^{2}}{1-\left| \rho_{l} \right|^{2}+\frac{\sigma^{2}_{\emph{n}}}{N_{\emph{T}}P}}.
\label{SINR-pro}
\end{equation}
\end{theorem}

\begin{IEEEproof}[Proof]
See Appendix \ref{proof_theorem4}.
\end{IEEEproof}

We now look into the high SNR regime. Recall from (20) that the performance of the proposed differential modulation scheme depends on the quality of two received signals. Because the second received symbol is subject to interference as shown in (19), the effective SINR settles as the transmit power increases, i.e., $\gamma_2 \rightarrow \frac{\left| \rho_{l} \right|^2}{1-\left| \rho_{l} \right|^2}$. Hence, we have
\begin{align}
\frac{2\gamma_{1}\gamma_{2}}{\gamma_{1}+\gamma_{2}} \approx 2\gamma_2\approx \frac{2\left| \rho_{l} \right|^2}{1-\left| \rho_{l} \right|^2}.
\end{align}
To this end, (\ref{PEP-pro}) reduces to
\begin{equation}
\begin{split}
&\textrm{Pr}\left \{ \textbf{G}\rightarrow {\textbf{G}}'|\textrm{SNR}\rightarrow \infty \right \}\leq\\
&\frac{1}{\left | \textbf{I}_{N_\emph{T}}+\frac{ \left| \rho_{l} \right|^{4}}{N_{\emph{T}}\left( 1+3\left| \rho_{l} \right|^{2} \right)\left( 1-\left| \rho_{l} \right|^{2} \right)  } \textbf{D}\left ( \textbf{G}-{\textbf{G}}' \right )\left ( \textbf{G}-{\textbf{G}}' \right )^{H}\textbf{D}^{H} \right |^{N_{\emph{R}}-l}},
\label{PEP-pro-highSNR}
\end{split}
\end{equation}
which indicates the existence of an error floor in the high SNR regime.

\begin{figure}[!t]\centering
\includegraphics[angle=0,scale=0.4]{Theorem2.eps}
\caption{SER induced by the proposed scheme versus the equivalent conventional one that exploits the signals received by antennas $\{ l+1,\dots,N_{\textrm{R}} \}$ at time $t_{i}$ and $\{ 1,\dots,N_{\textrm{R}}-l \}$ at time $t_{i+1}$ for differential decoding.}
\label{Fig_Theorem_2}
\end{figure}
\begin{figure}[!t]\centering
\includegraphics[angle=0,scale=0.4]{SNR.eps}
\caption{Impact of SNR on SER and PEP upper bound when $v=50,100$ and $150$m/s.}
\label{Fig_SERvsSNR}
\end{figure}

Fig. \ref{Fig_SERvsSNR} illustrates the SERs and PEP upper bounds under different SNR conditions when $v=\frac{mD}{\tau},~m=1,\dots,N_{\textrm{R}}-1$. It is intuitive that lower velocity and higher SNR lead to better error performance. As expected, an error floor is observed as the SNR becomes large. The red dashed line represents the PEP upper bound when $v=150$m/s as $\textrm{SNR}\rightarrow \infty$.

\section{Enhanced Scheme with Transmit Adaptation}
In Section III, we have analyzed the error performance of the proposed scheme with fixed $v$ and $M$. In this section, we show that by adaptively choosing the block length $M$ according to the velocity of the train, substantial performance gain can be obtained. To see this, we first take a look at the impact of block length on the error performance.

\subsection{Impact of Block Length}
According to (\ref{Correlation_case_a}), (\ref{Correlation_case_b}) and (\ref{alpha_and_beta2}), for a fixed $D$, we have
\begin{equation}
l=\left\{\begin{matrix}
\lfloor \frac{vMTt_{\textrm{s}}}{D} \rfloor,~~~~~~~~\textrm{for case I}
\\
\mathrm{Round}\left\{ \frac{vMTt_{\textrm{s}}}{D} \right\},~~\textrm{for case II}~~~
\end{matrix}\right.
\end{equation}
\begin{equation}
\rho_{l}=J_{0}\left( \frac{2\pi}{\lambda}\left( vMTt_{\textrm{s}} - lD \right) \right)e^{-c_{0}vMTt_{\textrm{s}} },
\end{equation}
and
\begin{equation}
\begin{split}
\rho_{l+1}=&
\left\{\begin{matrix}
J_{0}\left( \frac{2\pi}{\lambda}\left( vMTt_{\textrm{s}} - (l+1)D \right) \right)e^{-c_{0}vMTt_{\textrm{s}} },\\
\textrm{for case I}
\\
0,~~~~~~~~~~~~~~~~~~\textrm{for case II}
\end{matrix}\right.
\label{alpha_and_beta_and location}
\end{split}
\end{equation}
It can be observed that $M$ has a great impact on the values of $l$, $\rho_{l}$ and $\rho_{l+1}$. Meanwhile, an insightful observation is that, as the velocity $v$ changes, if the transmitter can adjust $M$ adaptively such that $vM$ stays approximately the same, then the channel correlation statistics $\textbf{C}$ shall not change significantly. Recall the PEP bounds given in (\ref{PEP}) and (\ref{PEP-pro}), we see that if the channel correlation pattern $\textbf{C}$ remains unchanged, then the corresponding error performance also stays the same. Therefore, adaptation of the block length $M$ leads to a much robust PEP and SER performance, which sheds light on how to combat the velocity variation.

\begin{figure}[!t]\centering
\includegraphics[angle=0,scale=0.4]{No_adaptation.eps}
\caption{Impact of block length $M$ on the SER performance.}
\label{Fig_SERvsTransInterval}
\end{figure}

We now illustrate the impact of block length on system error performance through the following example. For the simulation, we choose $M=1$, $2$, $3$, $4$, $5$ while other parameters remain the same as in Section III. From Fig. \ref{Fig_SERvsTransInterval}, we see that different block lengths result in distinct SER curves. Moreover, significant fluctuation in SER is observed when $M$ gets larger. This is rather intuitive, since a larger $M$ amplifies the influence of velocity variation on the channel correlation structure $\textbf{C}$, hence causing more frequent fluctuations. Furthermore, when $v=50$, $65$, $85$ and $125$m/s, by properly choosing $M$, i.e., $M = 5, 4, 3, 2$, the achievable SER remains the same, which corroborates the finding of Theorem \ref{theorem3}.

\subsection{Adaptive Transmission Method}
We now address the problem of how to adaptively set the block length $M$ as the velocity $v$ changes. Since the motivation of adaptation is to improve the error performance, we consider the optimal $M$ being the solution of the following optimization problem:
\begin{align*}
&\min_{M}\quad \textrm{PEP}_{\textrm{bound}}\left(v,M \right)\\
&~s.t.\quad M\in \mathbb{Z^{+}}
\end{align*}
where $\textrm{PEP}_{\textrm{bound}}$ follows (\ref{PEP}) for case I and (\ref{PEP-pro}) for case II. Since $M$ is a positive integer, we can obtain the optimal solution through exhaustive search (for clarity purposes, we denote this method as OptM). However, since there is no upper limit to the value of $M$, OptM may cause an intolerable overhead. To tackle this problem, we propose a much easier heuristic adaptation method, conserving computing resources while providing robustness.

Before presenting our method, let us revisit the numerical results in Fig. \ref{Fig_SERvsTransInterval}. When the train velocity is low (i.e., $v<50$m/s), we see $M=1$ provides the best SER performance. This is intuitive, since the successive channel samples are strongly correlated in time domain in this case. However, when the train velocity exceeds a certain threshold $v_{0}$ (i.e., $50$m/s), unit block length incurs a noticeable performance penalty due to the fact that channel's variation becomes increasingly significant. To address this issue, an intuitive solution is that the BS dynamically adjusts $M$ so that $vMTt_{\textrm{s}}\approx mD,~m=0,\dots,N_{\textrm{R}}-1$ (as depicted in Fig. \ref{Fig-matched-velocity}). As such, the influence of velocity variation is suppressed and, in the meantime, the level of spatial-temporal channel correlation will always be high enough to support differential decoding. As it is impossible to satisfy this criteria when $m=0$, we consider the case $m=1$, i.e., $vMTt_{\textrm{s}}\approx D$.

Now we are ready to present our heuristic transmit adaptation (HTA) method:

\emph{a)} In low $v$ regime (i.e., $v<v_{0}$), the BS exploits the proposed scheme with $M=1$.

\emph{b)} When $v$ exceeds the threshold $v_{0}$, the BS adaptively chooses $M$ according to $v$, so that $vMTt_{\textrm{s}}\approx D$. Given that $M\geq 1$, we have $M=\max \left\{ \mathrm{Round}\left\{ \frac{D}{vTt_{\textrm{s}}} \right\},1\right\}$.

\begin{figure}[!t]
\centering
\subfigure[Symbol error rates affected by different TA schemes.]{
\label{Fig-TA}
\includegraphics[angle=0,scale=0.4]{TX_adaptation.eps}}
\hspace{0.4in}
\subfigure[The discrete values of $M$ chosen by the BS.]{
\label{Fig-M}
\includegraphics[angle=0,scale=0.4]{M_Value.eps}}
\caption{SER performance and the corresponding $M$ induced by different adaptation methods with distinct antenna spacings.}
\label{Fig-TA-and-M}
\end{figure}

Now the remaining task lies in the threshold $v_{0}$. This is rather sample, since $v_{0}$ is the minimum solution of the following equation
\begin{equation}
\textrm{PEP}_{\textrm{bound}}\left( v,1\right)=\textrm{PEP}_{\textrm{bound}}\left( v,\mathrm{Round}\left\{ \frac{D}{vTt_{\textrm{s}}} \right\} \right).
\label{Velocity-Threshold}
\end{equation}

\newcounter{TempEqCnt3}
\setcounter{TempEqCnt3}{\value{equation}}
\setcounter{equation}{41}
\begin{figure*}[hb]
\hrulefill
\begin{equation}
\begin{split}
\widehat{\textbf{G}}
~=~&\mathrm{arg}\underset{\textbf{G}\in \mathcal{G}}{\mathrm{min}}\left \| \textbf{Y}_{i+1}-\textbf{C}\textbf{Y}_{i}\textbf{G} \right \|_{\textrm{F}}\\
~=~&\mathrm{arg}\underset{\textbf{G}\in \mathcal{G}}{\mathrm{min}}\textrm{Tr}\left\{ \textbf{Y}_{i+1}\textbf{Y}_{i+1}^{H}-\textbf{Y}_{i+1}\textbf{G}^{H}\textbf{Y}_{i}^{H}\textbf{C}^{H}-\textbf{C}\textbf{Y}_{i}\textbf{G}\textbf{Y}_{i+1}^{H}+\textbf{C}\textbf{Y}_{i}\textbf{G}\textbf{G}^{H}\textbf{Y}_{i}^{H}\textbf{C}^{H} \right\}\\
~=~&\mathrm{arg}\underset{\textbf{G}\in \mathcal{G}}{\mathrm{min}}\textrm{Tr}\left\{ \textbf{Y}_{i+1}\textbf{Y}_{i+1}^{H}-\textbf{Y}_{i+1}\textbf{G}^{H}\textbf{Y}_{i}^{H}\textbf{C}^{H}-\textbf{C}\textbf{Y}_{i}\textbf{G}\textbf{Y}_{i+1}^{H}+\textbf{C}\textbf{Y}_{i}\textbf{Y}_{i}^{H}\textbf{C}^{H} \right\}\\
~=~&\mathrm{arg}\underset{\textbf{G}\in \mathcal{G}}{\mathrm{max}}\textrm{Re}\left\{ \textrm{Tr} \left\{  \textbf{C}\textbf{Y}_{i}\textbf{G}\textbf{Y}_{i+1}^{H}   \right\}  \right\}
\label{Euclidean-detector}
\end{split}
\end{equation}
\end{figure*}
\setcounter{equation}{\value{TempEqCnt3}}

In Fig. \ref{Fig-TA-and-M}(a), we show numerically the error performance induced by different adaptation methods with distinct antenna spacings. For the case $D=0.05$m, invoking the non-adaptive scheme as the benchmark, we see that both OptM and HTA enhance the system robustness. It should be noted that the exhaustive-search based OptM scheme intends to minimize the PEP bound, not the real PEP or SER, hence it is possible that the OptM scheme performs worse than HTA for some particular choice of D = 0.05m. Yet, as shown in Fig. 10(a), except these particular cases, the OptM and HTA achieve almost the same SER performance. On this basis, it is reasonable to treat HTA as a near-optimal solution to the above optimization problem. Considering its low computational complexity, HTA may be a preferred option in high mobility settings. Moreover, for the proposed HTA scheme, it can be observed that setting $D=0.05$m outperforms the case $D=0.1$m when $v=110\sim 150$m/s. This observation indicates that, to fully exploit the spatial-temporal correlation, antenna spacing should be carefully designed if the statistical information (i.e., mean and variance) of the train velocity is known to the BS. Finally, Fig. \ref{Fig-TA-and-M}(b) depicts the corresponding values of $M$. When $D=0.05$m and $0.1$m, velocity thresholds $v_{0}$ are about $47$m/s and $48$m/s, respectively. Thereby, block lengths are both set to be $1$ when $v=0\sim 45$m/s. As $v$ exceeds $v_{0}$, a substantial increase in $M$ is observed. After that, $M$ will decrease gradually since $vMTt_s$ needs to be aligned to $D$. When it returns to $1$, HTA becomes of little avail because the train velocity is too high.

\section{Conclusion}
We have studied the joint spatial and temporal channel correlation inherent in high mobility scenarios with a moving antenna array. Capitalizing on this type of correlation information, an improved DSTM scheme was proposed, and the corresponding error performance was analyzed through the characterization of the PEP upper bound. Finally, an adaptive transmission scheme varying the block length according to the node velocity was proposed. The findings of the paper suggest that the proposed DSTM scheme outperforms the conventional scheme, and further performance improvement can be realized by adaptively adjusting the block length.

redpajama_set_name: RedPajamaArXiv


text:
Author: gstamper1317
Posted on January 17, 2022 by gstamper1317
Have you ever met a stranger and felt that you already knew them?
The story begins in the 1800s when Elias and Henry are souls on the Other Side. Their own time as humans lies ahead of them, but they unwittingly observe their ancestors-to-be as they progress through major historical events. From slave-liberation missions in antebellum America to the horror of the Vietnam War, they follow six generations across two centuries. But as the family separates, the descendants form two divergent lineages, thousands of miles apart, into which the two souls are eventually born.
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Somewhere Between Light and Time is a fast-paced adventure drama in which historical events and everyday reality are interwoven with elements of the supernatural. Aiden Leman lives in East Sussex, England, where he works in financial services for his day job. He studied International Relations in Scotland and was previously active in the teaching sector. He has worked in five different countries across three continents, although his dream job would probably be a firefighter.
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Publisher: Cecil Press
While trekking across South America, Ellie Bartlett finds a mysterious guidebook that changes the course of her journey—and her life.
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After losing out on yet another promotion, Ellie quits her job and buys a ticket to South America. Arriving at her hostel as a first-time backpacker, she finds a mysterious guidebook filled with cryptic messages about life and love. Intrigued, Ellie contacts the previous owner, Bella, who wrote her name under the heading THIS BOOK BELONGS TO.
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Ellie and Jerry start messaging, sharing their fears, hopes and desires. Following Bella's advice, Ellie embarks on a six-month solo adventure across South America, and over the course of emails, texts, phone calls and video chats, Ellie and Jerry's pen-pal friendship develops into something more…
But can you fall in love with someone you've never met?
Nick Levy loves traveling and writing to understand the world and our place in it. An award-winning scriptwriter and novelist, he has lived in dozens of countries, including Australia, the USA, Singapore, Mexico, Peru and Thailand. His passion for adventure has taken him from the highlands of Scotland to the coral reefs of Belize, and from the pyramids of Egypt to the tea plantations of Sri Lanka. With degrees in English, History and Education, he has taught in schools on three continents. His writing, directing and producing work in movies has crossed the genres of comedy, drama, thriller and action. Inspired by his parents' love of travel, film and books, he enjoys taking readers on a journey of discovery so they can appreciate our incredible planet and its fascinating inhabitants.
Posted on January 15, 2022 January 14, 2022 by gstamper1317
Political Historical Fiction
Date Published: September 29, 2021
Azad's father was brutally executed by the Iraqi army. Since then, Azad has moved to the city of Duhok to pursue his education. In doing so he hopes to provide a new life for his family and his fiancée, Juwan, who are back in their home village. But as the Iran-Iraq War comes to an end, the Iraqi government launches the genocidal Anfal campaign against the Kurds. Fleeing separately across physical and emotional borders, Azad and Juwan find themselves in the crosshairs of the Iraqi army. However, the most dangerous threat is the one they least expect.
Embrace Your Cause and Transform lives
Date Published: October 29, 2021
★★★ A roadmap like you've never seen before and a must read book to change your life and the lives of others ★★★
This book is a step by step guide book that reads as a thriller. It teaches you how to emulate the success Dr. Kteleh and his organization have had in holding the twenty-first century's greatest war criminal accountable, saving the lives of four million civilians, and protecting patient safety in the U.S. from the corporate medicine industry.
It also teaches how to:
Successfully advocate for a noble cause.
Create and utilize an effective network.
Build organizational capacity to support your goals
Effectively wield traditional and digital media to create public awareness
Meet the president of the United States and the most powerful policymakers at the heart of the nation's capital.
By following the step-by-step instructions based on real life experiences, you will be able to make progress advocating that you thought was never possible.
The only thing stopping you from making the world a better place is a lack of practical knowledge.
With this book in hand, you have no more excuses. Stop wishing and start advocating. Order your copy today!
The Secret to Finding Your Perfect
Date Published: Jan 4 2022
Publisher: Gatekeeper Press
Paid to be an exact size six under contract, a professional model, starting from her late teens to her late forties, shares how she has maintained her exact measurements for decades. In an informal, uninhibited voice she explains how she focuses on how she feels enabling her to eat whatever she wants, whenever she wants, while maintaining her body's natural size and how you can too!
This intimate guide encompasses all of her healthy lifestyle habits around nutrition, fitness, skincare, and wellness and includes insider secrets into the fashion world where she was paid to be "perfect."
Whether you're a size two, size 22, or any other size, her tips can help you find and maintain YOUR PERFECT!
This is an important read for girls and women. Coming from a model who has been through ups and downs with weight and weigh control.
It's important for us to understand our bodies, feel confident and comfortable in them and love ourselves no matter what our size.
This is a book that has important points, tips, and is a guide of sorts. Eye-opening and informative.
The Lion Trees @RABTBookTours @MKWebsiteandSEO
Date Published: August 25, 2014
Publisher:  OTF Literary, LLC
"THE LION TREES" — WINNER OF THE KINDLE BOOK AWARD!
What if survival required you to unlearn who you are? How far would you fall to save yourself? Sometimes happiness is a long way down.
The Johns family is unraveling. Hollis, a retired Ohio banker, isolates himself in esoteric hobbies and a dangerous flirtation with a colleague's daughter. Susan, his wife of forty years, risks everything for a second chance at who she might have been. David, their eldest, thrashes to stay afloat as his teaching career capsizes in a storm of accusations over a missing student and the legacy of Christopher Columbus. While Tilly, the black sheep, trades her literary promise for an improbable career as a starlet, and then struggles to define herself amidst a humiliating scandal and the judgment of an uncompromising writer.
By turns comical, suspenseful and poignant, the Johns family is tumbling toward the discovery that sometimes you have to let go of your identity to find out who you are.
Owen Thomas' rollicking debut novel is the winner of 16 international book awards, including: the 2015 Amazon Kindle Book Award for Literary Fiction, the 2015 Global eBook Award for New Adult Fiction, a 2015 Eric Hoffer Book Award, the 2015 'Book of The Year' for BooksAndAuthor.com, Finalist for the 2015 First Horizon Book Award, and placements at the London Book Festival, the New York Book Festival, the Amsterdam Book Festival, and the Beverly Hills International Book Awards.
Highly addictive, spectacular, and mind blowing… Thomas is a wizard of fiction. — U.S. Review of Books
A sweeping literary saga in the traditional 'Dr. Zhivago', 'Gone with the Wind', and 'The Thorn Birds', this book has it all… original and stirring… –The Eric Hoffer Book Award
[A] cerebral page turner…a powerful and promising debut.–Kirkus Reviews
Winner of 16 International Book Awards, including the Kindle Book Award, the Global eBook Award, The Eric Hoffer Book Award, the London Book Festival, the New York Book Festival, the Amsterdam Book Festival, and the Beverly Hills International Book Awards.
Fan Into Flame @RABTBookTours @MKWebsiteandSEO
Igniting A Heart for Christian Arsonism
Christian / Nonfiction / History
Publisher: MindStir Media
As the sun rose on the Twenty First Century, it was clear that something was wrong in the stuffy and traditional confines of the American Church: the church people within the building had lost connection with the community people outside of it. In so doing, they lost their burning relationship with Jesus Christ, who came from the very throne room of God to be in relationship with those alienated from Him.
In Fan Into Flame, we follow the relational story of Peter the Apostle with Jesus: from the simple spark ignited at their first meeting, through the trials by fire over three years, to the heart wrenching failure of Peter's denial. Throughout, the fisherman must step outside of his familiar surroundings with a bold willingness to have his weaknesses exposed, his sinfulness revealed, and his very life threatened. Peter must lose what he thought was important, gain what he could not obtain, and discover the commitment Jesus demands, the cost Jesus invites him to endure, and an incomprehensible love Jesus desires to give.
Like Peter, the church need not fear the unfamiliar or its own failures. It need only fear quenching the fire of the One who will never abandon her through both. This is the story of a true Christian Arsonist.
Joseph Skladany is an ordained pastor and has served in the Christian and Missionary Alliance since 2011.After pastoring in Ohio for three years, he moved to the Pacific Northwest where he is currently pastoring at Ignite Alliance in Veneta, Oregon. An out-of-the-church-box thinker, he has challenged the traditions of the often ingrown, static church, preferring to direct people's attentions to their community with an understanding of the current cultural context and armed with a sacrificial willingness to be open and fallible. Joseph has two adult children and his wife, Sandra, is a licensed counselor practicing in the Veneta, Eugene area.
Time of the Lioness: Internet Apocalypse – @sbredstone @RABTBookTours
Apocalypse, Post Apocalypse, Thriller, Female Heroine, New Adult
You're at work, in school, at home, out shopping, trapped in an elevator, or sky-high on an amusement park ride when the power and lights suddenly go out. For a moment, you're just annoyed, expecting only a momentary problem. Doesn't happen! Along with the power outage, all means of communication have been severed. The Internet is down, and a terrorist's face on the screen gloats. Fear now creeps into your optimism. Hours later, with a world overly
dependent on cyberspace, global economies begin to crumble from this irrevocable cyberattack. Totally freaked out, what will you do to survive the anarchy?
Annie Oakley Frye is a Cyber Warfare Specialist accepted cyberspace had become a cesspool of evil based on the multitude of daily cyberattacks. Annie does not live in a dystopian, fantasy world like Katniss Everdeen,
Beatrice Prior, and Lara Croft—she lives among us. Once she discovers this invisible menace is lurking in a million networks, and it can't be stopped from obliterating world economies, she sheds her mild-mannered personality
and becomes the lioness of her youth to survive. Out of kindness, she warns her friends about this approaching cyber nightmare, but they scoff at her insider information and doomsday scenario, believing the government won't
let it happen. Once the cyber nightmare launches a surprise arrives at her door!
"I can't put your book down. It's a page turner for sure. Really having an
awesome time reading it."
-Reviewer
If you are looking for a book that thinks outside of the box and really makes your mind work overtime, this is it.
We get a female heroine in Annie who may initially seem meek, but really comes out of her shell to face things head on. I loved her growth in this book!
I definitely think this is one of those dive in and don't get up kind of books. It will hook you as you get lost in the words.
Like Shakespeare and Hemmingway, I am an eclectic writer. My stories
develop from my imagination and passions, usually just before I fall asleep.
Before I became an Indie author, I had two novels published by a respected
independent publisher, which went out of business. Since books cannot be
removed from Amazon, A Sinister Obsession is now Dead Can't Haunt You, and
Stardust Dreams will soon be Forrester's Folly. During my clinical careers
as a School Psychologist and Clinical Social Worker in private practice, I
had a book on human nature published by a traditional publisher and when the
editor passed away, it vanished. That book will be updated and launched next
year. Happily, I am now an Indie author with the same quality of work but
now under my control. My genres have been horror, suspense/thriller, senior
romance, sci/fi, and now apocalyptic, which is based on my worries about the
precarious future of cyberspace. Besides writing, I am an avid golfer,
bowler, and tennis player. I reside in Florida with my lovely wife and
Mickey, our handsome Shih Tzu.
A Touch of Romance – @paullettgolden – @RABTBookTours – @BookBuzznet
Romantic Encounters, Book 2
Despite the odds, two poets manifest their dreams.
His recent appointment as Regius Professor has Jules Knowlton enjoying a life of leisure, one that begins with a writing retreat to stimulate his mind but ends with the unanticipated arousal of his heart. The problem? His newfound ladylove is arranged to marry someone else.
Leila Owen is a loyal and devoted daughter, one who would do anything for her family, anything except marry the suitor they've chosen for her. Using her poetic cunning, she schemes ways to convince her parents the man of her choice is a better match. The problem? The suitor is nothing if not perfect.
This is the love story of Jules and Leila, two poets who playact through life's charade.
From second chance romances to mistaken identities, experience A Touch of Romance in this collection of one short novel and twelve bonus flash fiction pieces.
Other Books in the Romantic Encounters series:
A Dash of Romance
With quills and fantasies, they write their future.
Percival Randall lives an uncomplicated life of luxury, as he likes it. Not even an ultimatum requiring marriage can tip the scales. A conniving young lady who compromises his name to force a betrothal, however, is an impediment to happiness Percival must confront.
Abigail Walsley dreams of publishing novels rather than marrying dashing heroes. An unexpected proposal and a subsequent Banbury tale tumble her into a betrothal with a man she has never met. Following her dreams proves a challenge with a marriage of inconvenience on the horizon.
This is the love story of Percival and Abigail, two dreamers who write their love story one scene at a time.
From second chance romances to mistaken identities, experience A Dash of Romance in this collection of one short novel and fourteen bonus flash fiction pieces.
Her name resonated in a deep voice; the sound more akin to the reverberation of a gong than spoken word. A thrill tickled down her spine.
Eyes forward, she met the gaze of her partner.
If her shoulders sagged with disappointment, she hoped no one noticed. The gentleman who held the bit of paper with her name was a tidy sort of man. Average height, average build, short-cropped brown hair, staid expression, and, oh goodness, spectacles perched on the bridge of his nose. He was exactly the sort of gentleman she might expect to find at a writers' retreat.
Leila sighed. It would be a long afternoon.
He took his counterpart's measure. Her countenance spoke of someone who would rather be anywhere except by his side. The subtle assessment he made of her person was for scholarly purposes.
The length of her eyelashes suggested she was, how should he say, flighty. The straightness of her nose implied a confidence bordering on conceit. The fullness of her lips invited untoward attention. By a connoisseur of women, she would be considered a stunning beauty with her soulful brown eyes, sleek ebony hair, and bronze skin denoting a South Asian heritage, but all Jules recognized was the arrogance of her carriage. She would not be his preferred partner for a treasure hunt.
He sighed. It would be a long afternoon.
Celebrated for her complex characters, realistic conflicts, and sensual love scenes, Paullett Golden puts a spin on historical romance. Her novels, set primarily in Georgian and Regency England with some dabbling in Ireland, Scotland, and France, challenge the norm by involving characters who are loved for their flaws, imperfections, and idiosyncrasies. Her stories show love overcoming adversity. Whatever our self-doubts, love will out.
By day, Paullett is a university professor, teaching argumentation, composition & rhetoric, technical/business writing, creative writing, and literature. She enjoys spending time in the places that inspire her stories. While some travel is for work, most travel is for story research to get a sense of the land, the culture, the people, and the history.
As an ovarian cancer survivor, she takes pleasure in every moment. Her hobbies beyond writing include butterfly and pollinator gardening, orchid rescuing, exotic plant collecting, classic car showing, drag racing, motorbike cruising, powerlifting & bodybuilding, music performing, and feathered and furry children raising.
BookBuzz
Salvo – @RABTBookTours – @MKWebsiteandSEO
Nonfiction / Religion
This book was written in the state of breathless Samadhi, on a cell phone. Pithy mystical aphorisms began arriving from "ourselves in the future". These future selves have survived the cataclysm we are currently undergoing and want to extend a hand to help us through the Apocalyptic labyrinth safely to the other side. This journey is not what is expected. The price of admission to the New Earth turns out to be realization of every terror we've ever imagined. Turns out, it was only ourselves we were afraid of in the first place! The final reward is the acceptance and embracing of this glorious terrifying Self. What we were looking for is what is looking for us.
Randy was born in the Blue Ridge Mountains of Virginia. After graduating summa cum laude from Virginia Tech and receiving a master's degree in Family Therapy, he completed his doctoral degree at the Institute of Transpersonal Psychology in Palo Alto, CA. By far, most of his education was delivered by the "School of Hard Knocks." His gift is in seeing the miracle in everything, even in a tragedy — even in the current global catastrophe. He is now back in the Blue Ridge Mountains, in the house he grew up in, where he is looking forward to growing old and dying. Or not.

redpajama_set_name: RedPajamaCommonCrawl


← Fabulous WEEKEND WEATHER Ahead!
So great to see the sunshine today. Yes, that bright object in the sky that has been hiding behind the clouds. It made a grand appearance this morning.
There will be several reasons to look up over the next several days especially in the night sky.Tonight the spring star Arcturus will be making an appearance. It will be visible the rest of February and evenings in March. It will climb in the east sky. To find it look for the Big Dipper then follow the imaginary handle. It will point you to the star.
Saturday look east during the early evening, that brilliant object you will see is no star. It's Jupiter just above the horizon. It will reach it's highest point about 12midnight. Jupiter can guide you to the Beehive star cluster over the next few months. Look for the star Regulus above Jupiter in the early evening. When you find it draw an imaginary line through Jupiter to Regulus, keep going and you should see the Beehive cluster.
Sunday night there will be an increase in clouds and a chance of rain so this next one may be tricky to see. You will be looking for the red planet and the waning gibbous moon. You can stay up late around midnight or get up early before dawn to see it. Predawn on Monday is better because Mars will be higher in the sky.
On Tuesday morning (March 1) before the sun comes up you can see the moon, Mars and Saturn. To find the planets, the lit side of the moon will point toward Saturn and the dark side is in the direction of Mars. They will look different. Saturn appears golden and Mars looks reddish. Binoculars will help you see them better.
The International Space Station will also be passing over next week. Some days will be better than others for viewing because of the weather and also because of the maximum height. Here's the list.
I hope you'll be able to see some of these sights in the sky. Thanks to earthsky.com for the great graphics and information.


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Dortmund  (; en bajo alemán: Düörpm; en latín: Tremonia, nombre hispánico tradicional) es una ciudad alemana localizada en el estado federado de Renania del Norte-Westfalia, está situada en la Región del Ruhr, en el área metropolitana del Rin-Ruhr y es la octava más grande de Alemania, con una población de  habitantes (a finales de 2019). Es considerada el centro administrativo, comercial y cultural del Ruhr oriental. El río Ruhr fluye al sur de la ciudad, y el pequeño río Emscher atraviesa el término municipal. El canal Dortmund-Ems también acaba en el puerto de Dortmund, que es el mayor puerto de canal europeo, y tiene enlaces de Dortmund hasta el mar del Norte. Dortmund se conoce como la "metrópoli verde" de Westfalia.

Fundada alrededor del año 882, Dortmund se convirtió en una Ciudad Libre Imperial. A lo largo de los siglos  y  fue la "ciudad principal" del Rin, Westfalia y el Círculo Holandés de la Liga Hanseática. Después de la Guerra de los Treinta Años, la ciudad fue destruida y su importancia disminuyó hasta el inicio de la industrialización. La ciudad se convirtió entonces en uno de los centros de producción de carbón, acero y cerveza más importantes de Alemania. Dortmund fue una de las ciudades más bombardeadas de Alemania durante la Segunda Guerra Mundial. Los devastadores bombardeos del 12 de marzo de 1945 destruyeron el 98 % de los edificios del centro de la ciudad. Estos bombardeos, con más de 1110 aviones, mantienen el récord de un solo objetivo en la Segunda Guerra Mundial.

La región se ha adaptado desde el colapso de sus industrias siderúrgicas y carboníferas centenarias y ha pasado a la tecnología biomédica de alta tecnología, la tecnología de microsistemas y también los servicios. En 2009, Dortmund fue clasificada como ciudad Nodo en el Índice de Ciudades Innovadoras publicado por 2thinknow; es la ciudad más sostenible y digital de Alemania. Hoy en día, Dortmund es la sede de numerosas instituciones culturales y educativas, incluyendo la Universidad Técnica de Dortmund y la Universidad de Ciencias Aplicadas y Artes de Dortmund, la Escuela Internacional de Administración y otras instalaciones educativas, culturales y administrativas con más de 49 000 estudiantes.

Historia
A comienzos del , como consecuencia de las guerras Napoleónicas, el territorio que comprende este estado pasó a formar parte de la Confederación del Rin.

Durante el Primer Imperio francés la parte de la región al oeste del río Rin fue designada como Departamento del Roer. En 1807 Jerónimo Bonaparte es nombrado rey de Westfalia.

Tras la Segunda Guerra Mundial el estado de Renania del Norte-Westfalia fue establecido por iniciativa de la ocupación militar británica el 24 de agosto de 1946. Originalmente consistía en Westfalia y el norte de la provincia del Rin, que antes pertenecieron a Prusia. En 1947 el antiguo estado de Lippe se fusionó con el de Renania del Norte-Westfalia, lo que condujo a las fronteras actuales del Estado. A continuación, fue ratificado mediante un plebiscito la Constitución de Renania del Norte-Westfalia. A diferencia de otros estados alemanes Renania del Norte-Westfalia no tenía antecesores históricos. La atención se centró sobre todo en el deseo de los aliados para integrar un territorio común, en la rica región del Ruhr.

Encontrar una identidad común de Lippe, Westfalia y Renania fue un gran desafío en los primeros años del Land. Los mayores retos en la postguerra fueron la reconstrucción y el establecimiento de un Estado democrático. A continuación, debió rediseñar la estructura económica desarrollada como resultado del declive de la industria minera que fue tema central de la política nacional.

Las elecciones de Renania del Norte-Westfalia del 22 de mayo de 2005 concedieron a la CDU una victoria inesperada. Su principal candidato Jürgen Rüttgers construyó un nuevo gobierno de coalición formado por la CDU y el FDP que sustituyó al anterior gobierno encabezado por Peer Steinbrück. Rüttgers fue elegido nuevo primer ministro (en alemán Ministerpräsident) del estado federal el 22 de junio de 2005.

Entre 1905 y 1979 Dortmund ha incorporado a varios municipios; por ejemplo Körne, Deusen, Eving, Huckarde, Hörde, Aplerbeck y Holzen.

Geografía

Limita: al norte, este y sureste con Unna (distrito) (al norte con la ciudad de Lünen, al este con las ciudades de Kamen y Unna, al sureste con la municipalidad de Holzwickede y la ciudad de Schwerte), al sur con la ciudad libre de Hagen, al suroeste con el distrito de Ennepe-Ruhr, al oeste con la ciudad libre de Bochum, y al noroeste con el distrito de Recklinghausen (ciudades de Herten, Recklinghausen y Castrop-Rauxel).

Código Postal: 44001 hasta 44388. Código telefónico 0231.

Clima
Dortmund está situado en la zona de clima templado. Los inviernos son ligeramente fríos, los veranos más bien frescos. La temperatura media anual se encuentra en 8 °C. El total de la cantidad media anual de precipitación es de 800 mm. Precipitación uniforme cae durante todo el año, la lluvia constante que prevalece en el invierno, chubascos aislados dominan la temporada de verano. Dortmund tiene características de zonas densamente pobladas, como es típico la aparición de islas de calor urbano.

Demografía

En Dortmund la población creció rápidamente en tiempos de la industrialización del , cuando comenzó el carbón, la minería y el procesamiento de acero. Por primera vez en 1904 más de 100 000 personas vivían en Dortmund. Las cifras de población habían aumentado constantemente hasta los 657 804 en 1965.

Posteriormente, las cifras de población han caído a aproximadamente 580 000 en 2011; las proyecciones prevén un descenso a 550 000 habitantes en 2030. A diferencia de las proyecciones de las cifras de población han sido ligeramente ascendente en los años anteriores, que es debido a los beneficios netos de migración. Especialmente personas más jóvenes (18 a 25 años de edad) llegan a establecerse en Dortmund principalmente por sus universidades u otras actividades relacionadas con la educación. A partir de 2012, Dortmund tenía una población de 579 012 habitantes de los cuales unos 177 000 (30 %) eran de origen no alemán.

Economía

Dortmund se ha adaptado desde el colapso de sus industrias siderúrgicas, del carbón y de la cerveza, que han durado un siglo. La región se ha pasado a la alta tecnología, la robótica, la tecnología biomédica, la tecnología de microsistemas, la ingeniería, el turismo, las finanzas, la educación y los servicios, por lo que es una de las ciudades de nueva economía más dinámicas de Alemania. En 2009, Dortmund fue clasificada como ciudad Nodo en el Índice de Ciudades Innovadoras publicado por 2thinknow. En 2012, las cámaras de comercio de Zaragoza y Dortmund firmaron un acuerdo para impulsar la colaboración ente pymes aragonesas y alemanasDortmund alberga cierto número de pequeñas y medianas empresas de tecnología de la información, muchas de ellas vinculadas al programa local de la universidad TechnologieZentrumDortmund. La ciudad trabaja en estrecha colaboración con los institutos de investigación, las universidades privadas y las empresas para colaborar en las iniciativas de comercialización de la ciencia.

Dortmund es famosa por su cerveza (Export) y su club de fútbol Borussia Dortmund (BVB 09).

Turismo

El turismo en Dortmund es un factor económico de rápido crecimiento cada año: se pueden anunciar nuevos récords de pernoctaciones, se abren nuevos hoteles y se añaden nuevos imanes para los visitantes. A partir de mediados de la década de 1990, Dortmund, que antes era un centro industrial, experimentó un rápido desarrollo que amplió sus posibilidades culturales y turísticas, y la transformó en una ciudad recientemente vibrante. Un paso estratégico importante fue el inicio de la construcción del nuevo Konzerthaus Dortmund, la reutilización de viejos edificios industriales vacíos como el Zollern II/IV Colliery, Kokerei Hansa, Dortmund U-Tower y la reorientación estratégica del mercado navideño de Dortmund, con más de 300 puestos en torno a una gigantesca creación de árboles de Navidad de 45 m de altura, considerados los más grandes del mundo. Un nuevo centro de información turística justo al lado de la Torre en U, ofrece a los visitantes una rápida visión general de las atracciones turísticas de la ciudad y de la zona del Ruhr. Hoy en día, Dortmund es, con más de 1 450 528 (2017) pernoctaciones, uno de los destinos más populares de Renania del Norte-Westfalia.

La mayoría de los turistas son nacionales, procedentes de Alemania. Los viajeros internacionales llegan desde el Reino Unido, los Países Bajos, Austria y Suiza.

Transporte

El aeropuerto de Dortmund se encuentra a 13 km al este del centro, muy próximo a la ciudad de Holzwickede. Próximo a Dortmund también está el aeropuerto internacional de Düsseldorf, tercero de Alemania por número de pasajeros solo por detrás de los de Múnich y Fráncfort del Meno. Desde el año 2000 el aeropuerto de Dortmund ha ido experimentando un incremento del tráfico aéreo gracias a las compañías de bajo coste.

Transporte urbano
La principal arteria de transporte público en Dortmund es una red extensa de autobuses y una red de Stadtbahn (metro ligero con 20,5 km en túneles y 27 estaciones subterráneas)) e incluso tranvías.

Tren rápido

El S-Bahn Rin-Ruhr es un tren rápido que cubre la macro-región metropilitana alemana del Rin-Ruhr. Desde la ciudades de Bonn y Colonia, hasta la ciudad de Dortmund.

Cultura
La ciudad tiene una larga tradición en la música y el teatro. La orquesta fue fundada en 1887 y ahora se llama Dortmunder Philharmoniker. La primera casa de ópera fue construida en 1904, destruida en la Segunda Guerra Mundial y se abrió de nuevo en 1966 como Opernhaus Dortmund. Es operada por el teatro Dortmund junto con otros lugares, incluyendo (desde 2002), el Konzerthaus de Dortmund. El Dortmund U-Tower, que una vez fue una fábrica de cerveza, es ahora el centro de las industrias creativas y el Ostwall Museum am. La ciudad toma su nombre de la cerveza Dortmunder estilo y es la sede de la fábrica de cerveza Dortmunder actio.

Administración
Dortmund está estructurado en doce distritos municipales:

Deportes

Borussia Dortmund es el club de fútbol de la ciudad. Compite en la Bundesliga, la primera división del fútbol nacional. Disputa sus encuentros de local en el Estadio Signal Iduna Park, que posee un aforo superior a 81 000 espectadores.

Ciudades hermanadas
Dortmund está hermanado con:

Personas notables

Referencias

Enlaces externos

 Ayuntamiento de Dortmund

Ciudades de la Liga Hanseática
Localidades de Renania del Norte-Westfalia

redpajama_set_name: RedPajamaWikipedia


Alexander Aksenyonok is Ambassador Extraordinary and Plenipotentiary of the Russian Federation, a veteran diplomat and Arabist who has had assignments to many Arab countries. He served as Ambassador to Algeria and Slovakia, and as Special Envoy to the Balkans. He has a Doctorate in Law.
The present global interdependence makes any conflict take quite new, hitherto unknown shapes; so it is simply impossible to predict how events will develop if one simulates them on the basis of the experience of the "first" Cold War.
Can Kosovo's imminent independence be seen as a precedent in resolving other internal state conflicts, including in the former Soviet republics? This problem has gone from being a subject of academic dispute to an explosive element of Realpolitik.
A simplified approach to the Middle East problems does not conform to local realities. It would be more fruitful to follow the 'do-no-harm' principle, separating what must be reformed from traditional elements of life that do not impede the modernization processes.


text:
Lobbyists didn't suffer a slowdown in 2009
Health industry spent $544m
By Jonathan D. Salant
Bloomberg News / February 13, 2010
WASHINGTON - The recession did little to slow lobbyists in the nation's capital last year, the Center for Responsive Politics reported yesterday as it detailed a 5 percent growth in expenses to a record $3.47 billion.
The US Chamber of Commerce led the way, spending a record $144 million. No group had eclipsed the $100 million threshold. Exxon Mobil, based in Irving, Texas, was next with $27.4 million.
In 2008, the Chamber of Commerce spent $91.7 million.
"Lobbying appears recession-proof,'' Sheila Krumholz, executive director of the Center for Responsive Politics research group, said in a statement. "Even when companies are scaling back other operations, many view lobbying as a critical tool in protecting their future interests.''
General Electric, based in Fairfield, Conn., was the second biggest-spending corporation with $25.5 million, up from $19.4 million in 2008. GE's NBC Universal subsidiary is seeking approval to merge with Philadelphia-based Comcast.
The health industry spent $544 million, up 12 percent over 2008, as Congress debated legislation to overhaul health care. The Senate and House have passed separate versions of a health care plan.
Final health care legislation has been held up as Democrats decide how to proceed following the Senate victory of Republican Scott Brown in Massachusetts. That cost the Democrats the 60th vote they needed to break Republican filibusters.
The drug industry's trade group, Pharmaceutical Research and Manufacturers of America, spent $26.2 million, 30 percent higher than 2008 and third-most among individual concerns hiring lobbyists in 2009.
Four other health care groups were among the top 10 spenders last year: New York-based Pfizer, which spent $24.6 million; Chicago-based Blue Cross and Blue Shield Association and its members, $22.7 million; Washington-based AARP, formerly the American Association of Retired Persons, $21 million; and the Chicago-based American Medical Association, $20.8 million.
The US Chamber of Commerce also spent heavily on the health care overhaul. Some of the chamber's overall spending included "grass-roots'' lobbying efforts, which most other organizations don't disclose, the center noted.
A nonprofit, independent group, the Center for Responsive Politics tracks money in US politics and its effect on elections and public policy.
It reported that the finance, insurance, and real estate industry, facing congressional proposals for stronger regulation, spent $465 million, up 1 percent from 2008. Some large financial institutions, including Bank of America and New York-based Goldman Sachs, decreased their lobbying spending as they accepted federal bailout funds under the Troubled Asset Relief Program.
Other previous lobbying stalwarts had a more precipitous decline. American International Group, for instance, spent just $2.27 million on federal lobbying - a quarter of its 2008 spending - before shutting down lobbying operations in June. Mortgage giants Fannie Mae and Freddie Mac stopped lobbying after each spent tens of millions of dollars earlier in the decade.
The number of companies or entities that reported lobbying the federal government in 2009 increased to 15,712, from 15,049. But the number of registered federal lobbyists decreased, falling to 13,742 from 14,442 in 2008. That did not keep the overall expenditures from accelerating. Spending, fueled by intensifying efforts to overhaul health care and financial regulations in the fall, reached $955.1 million in the last quarter, the first quarter in history that expenditures cracked the $900 million mark.
"Despite the odds, last year was a record year for lobbying,'' Krumholz said. "However, it's entirely possible that even more lobbying dollars will be spent in 2010.''
Top lobbying associations
Pharmaceutical and health products: $266.8 million
General business associations: $183 million
Oil and gas: $168.4 million
Insurance: $164.2 million
Electric utilities: $144.4 million
Computer/Internet companies: $118.9 million
General manufacturing and distributing: $113.4 million
Hospitals and nursing homes: $108.4 million
Television, movies, and music: $107.3 million
Education: $98.6 million
Source: Center for Responsive Politics
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