text:
\section{Introduction}\label{secI}

We consider a three-terminal state-dependent relay channel (RC) in which, as shown in Figure~\ref{StateDependentDiscreteMemorylessRelayChannel}, the source wants to communicate a message $W$ to the destination through the state-dependent RC in $n$ uses of the channel, with the help of the relay. The channel outputs $Y_2$ and $Y_3$ for  the relay and the destination, respectively, are controlled by the channel input $X_1$, the relay input $X_2$ and the channel state $S$, through a given memoryless probability law $W_{Y_2,Y_3|X_1,X_2,S}$. The channel state $S$ is generated according to  a given memoryless probability law $Q_S$. It is assumed that the channel state is known, noncausally, to only the source. The destination estimates the message sent by the source from the received channel output. In this paper we study the capacity of this communication system. We refer to this model as \textit{state-dependent RC with informed source}.

\begin{figure}[htpb]
\centering
        \includegraphics[width=\linewidth]{StateDependentRelayChannelWithInformedSource.pdf}
\caption{Relay channel with state information $S^n$ available non-causally at only the source.}
\label{StateDependentDiscreteMemorylessRelayChannel}
\end{figure}

The state-dependent multiaccess channel (MAC) with only one informed encoder and degraded message sets is considered in \cite{KL07a,SBSV07a}; and the state-dependent relay channel (RC) with only informed relay is considered in \cite{ZKLV08b}. For all these models, the authors develop non-trivial outer or upper bounds that permit to characterize the rate loss due to not knowing the state at the uninformed encoders. Key feature to the development of these outer or upper bounds is that, in all these models, the uninformed encoder not only does not know the channel state but, also, can learn no information about it.

The model for the RC with informed source that we study in this paper seemingly exhibits some similarities with the RC with informed relay considered in \cite{ZKLV08b}, and it also connects with the MAC with asymmetric CSI and degraded message sets considered in \cite{KL07a,SBSV07a,ZKLV09a} and with the MAC with two states considered in \cite{PKEZ07}. However, establishing a non-trivial upper bound for the present model is more involved, comparatively. Partly, this is because, here, the uninformed encoder is also a receiver; and, so, it can potentially get some information about the channel states from directly observing its output. That is, at time $i$, the input $X_{2,i}$ of the relay can potentially depend on the channel states through $Y_2^{i-1}=(Y_{2,1},\hdots,Y_{2,i-1})$. Further, since, for $j=1,\hdots,i-1$, $Y_2^j$ may depend on the channel states in a non-causal manner (through the source codeword $X_{1,j}(W,S^n)$), and not only through the current state $S_i$, so does the input of the relay, potentially.

Establishing good lower bounds for the present model is also a non-easy task, due to the asymmetry caused by knowing the state at only the source. In this paper, we establish two lower bounds on the capacity of the state-dependent RC with informed source, for both discrete memoryless (DM) and memoryless Gaussian cases. For the Gaussian case, we also establish a non-trivial upper bound that is strictly better than the max-flow min cut or cut-set upper bound. Our lower bounds exploit ideas that are inherently different; and, so, their comparison helps providing right guidance towards the appropriate design. We obtain the first lower bound by a coding scheme in which the source describes the known state to the relay and destination \textit{ahead of time}, in addition to sending the information message. The relay performs collaborative binning against the learned state, through a combined binning and decode-and-forward (DF) scheme. We obtain the second lower bound by a coding scheme in which, rather than the channel state itself, the source describes to the relay the appropriate input that the relay would send had the relay known the channel state. The relay then simply guesses this input and sends it in the appropriate block. The lower bound obtained with this scheme achieves close to optimal for some special cases.

We note that the lower bounds that we develop in this paper are tailored to primarily overcome the state asymmetry; and, so, they perform well in the situations in which classic DF for RC without state is suitable. Other interesting achievability results which perform well in other situations can be found in \cite{AMA09,KS09,ZV09b}.

\section{System Model and Definitions}\label{secII}
As shown in Figure~\ref{StateDependentDiscreteMemorylessRelayChannel}, we consider a state-dependent relay channel  denoted by $W_{Y_2,Y_3|X_1,X_2,S}$ whose outputs $Y_2 \in \mc Y_2$ and $Y_3 \in \mc Y_3$ for the relay and the destination, respectively, are controlled by the channel inputs $X_1 \in \mc X_1$ from the source and $X_2 \in \mc X_2$ from the relay, along with a random state parameter $S \in \mc S$. It is assumed that the channel state $S_i$ at time instant $i$ is independently drawn from a given distribution $Q_S$ and the channel states $S^n$ are non-causally known at the source.

The source wants to transmit a message $W$ to the destination with the help of the relay, in $n$ channel uses. The message $W$ is assumed to be uniformly distributed over the set $\mc W=\{1,\hdots,M\}$. The information rate $R$ is defined as $n^{-1}\log M$ bits per transmission.

\noindent An $(M,n)$ code for the state-dependent relay channel with informed source consists of an encoding function at the source
$$\phi_1^n:~~\{1,\hdots,M\}\times {\mathcal{S}}^n \rightarrow \mc X_1^n,$$ a sequence of encoding functions at the relay
$$\phi_{2,i}:~~ {\mathcal{Y}}_{2,1}^{i-1} \rightarrow {\mathcal{X}}_2,$$
for $i=1,2,\ldots,n,$ and a decoding function at the destination
$$\psi^n: ~~ \mc Y_3^n \rightarrow \{1,\hdots,M\}.$$

Let a $(M,n)$ code be given. The sequences $X_{1}^n$ and $X_{2}^n$ from the source and the relay, respectively, are transmitted across a state-dependent relay channel modeled as a memoryless conditional probability distribution $W_{Y_2,Y_3|X_1,X_2,S}$. The joint probability mass function on ${\mc W}{\times}{\mc S^n}{\times}{\mc X^n_1}{\times}{\mc X^n_2}{\times}{\mc Y^n_2}{\times}{\mc Y^n_3}$ is given by
\begin{align}
P(w,s^n,x^n_1,x^n_2,y^n_2,y^n_3) = &P(w)\prod_{i=1}^{n}Q_S(s_i)P(x_{1,i}|w,s^n)P(x_{2,i}|y^{i-1}_2)\nonumber\\
&{\cdot}W_{Y_2,Y_3|X_1,X_2,S}(y_{2,i},y_{3,i}|x_{1,i},x_{2,i},s_i).
\end{align}

The destination estimates the message sent by the source from the channel output $Y_{3}^n$. The average probability of error is defined as $P_e^n = \mathbb{E}_{S}\big[\mathrm{Pr}\big(\psi^n(Y_{3}^n)\neq W|S^n=s^n\big)\big].$

An  $(\epsilon,n,R)$ code for the state-dependent RC with informed source is an $(2^{nR},n)-$code $(\phi_1^n,\phi_2^n,\psi^n)$ having average probability of error $P_e^n$ not exceeding $\epsilon$.

A rate $R$ is said to be achievable if there exists a sequence of $(\epsilon_n,n,R)-$codes with $\lim_{n \rightarrow \infty} \epsilon_n=0$. The capacity $\mc C$ of the state-dependent RC with informed source is defined as the supremum of the set of achievable rates.

Due to space limitation, the results of this paper are either outlined only or mentioned without proofs. Detailed proofs can be found in \cite{ZS10}.

\section{The Discrete Memoryless RC with Informed Source}\label{secIII}

In this section, we assume that the alphabets ${\mathcal{S}}$, ${\mathcal{X}}_1$, ${\mathcal{X}}_2$, ${\mathcal{Y}}_2$, ${\mathcal{Y}}_3$ in the model are all discrete and finite.

\vspace{0.1cm}

\begin{theorem}\label{Theorem__LowerBound1__DiscreteChannel}
The capacity of the discrete memoryless state-dependent relay channel with informed source is lower bounded by
\begin{align}
R^{\text{lo}} = \max \: \min\: \{&I(U;Y_2|V,\hat{S}_R)-I(U;S,\hat{S}_D|V,\hat{S}_R),\nonumber\\
&I(U,V;Y_3|\hat{S}_D)-I(U,V;S,\hat{S}_R|\hat{S}_D)\}
\label{AchievableRate__Theorem__LowerBound1__DiscreteChannel}
\end{align}
subject to the constraints
\begin{subequations}
\begin{align}
I(S;\hat{S}_R) &\leq I(U_R;Y_2,\hat{S}_R|U,V)-I(U_R;S,\hat{S}_R,\hat{S}_D|U,V)\\
I(S;\hat{S}_D) &\leq I(U_D;Y_3,\hat{S}_D|U,V)-I(U_D;S,\hat{S}_R,\hat{S}_D|U,V)\nonumber\\
&+[I(U;Y_3,\hat{S}_D|V)-I(U;S,\hat{S}_R,\hat{S}_D|V)]_{-}\\
I(S;\hat{S}_R,\hat{S}_D) + I(\hat{S}_R;\hat{S}_D) &\leq I(U_R;Y_2,\hat{S}_R|U,V)-I(U_R;S,\hat{S}_R,\hat{S}_D|U,V)\nonumber\\
&+I(U_D;Y_3,\hat{S}_D|U,V)-I(U_D;S,\hat{S}_R,\hat{S}_D|U,V)\nonumber\\
&+[I(U;Y_3,\hat{S}_D|V)-I(U;S,\hat{S}_R,\hat{S}_D|V)]_{-}\nonumber\\
&-I(U_R;U_D|U,V,S,\hat{S}_R,\hat{S}_D)
\label{Constraints__AchievableRate__Theorem__LowerBound1__DiscreteChannel}
\end{align}
\end{subequations}
where $[x]_{-} \triangleq \min(x,0)$, and the maximization is over all joint measures on ${\mc S}\times\hat{\mc S}_R\times\hat{\mc S}_D\times{\mc U}_R\times{\mc U}_D\times{\mc U}\times{\mc V}\times{\mc X}_1\times{\mc X}_2\times{\mc Y}_2\times{\mc Y}_3$ of the form
\begin{align}
&P_{S,\hat{S}_R,\hat{S}_D,U_R,U_D,U,V,X_1,X_2,Y_2,Y_3} = Q_SP_{\hat{S}_R,\hat{S}_D|S}P_{V|\hat{S}_R}P_{U|V,S,\hat{S}_R,\hat{S}_D}P_{U_R,U_D|V,U,S,\hat{S}_R,\hat{S}_D}\nonumber\\
&\qquad P_{X_1|U_R,U_D,U,V,S,\hat{S}_R,\hat{S}_D}P_{X_2|V,\hat{S}_R}W_{Y_2,Y_3|X_1,X_2,S}.
\label{Measure__AchievableRate__Theorem__LowerBound1__DiscreteChannel}
\end{align}
and satisfying
\begin{align}
I(V;Y_3,\hat{S}_D) - I(V;\hat{S}_R) > 0.
\end{align}
\end{theorem}
\vspace{0.3cm}

\begin{remark}\label{remark1}
The intuition for the coding scheme which we use to establish the lower bound in Theorem~\ref{Theorem__LowerBound1__DiscreteChannel} is as follows. Had the relay known the state, the source and the relay could implement collaborative binning against that state for transmission to the destination \cite{KSS04}. Since the source knows the state of the channel non-causally, it can transmit a description of it to the relay \textit{ahead of time}. The relay recovers the state (with a certain distortion), and  then utilizes it in the relevant block through a collaborative binning scheme. The hope is that the benefit that the source can get from being assisted by a more capable relay largely compensates the loss caused by the source's spending some of its resources to make the relay learn the state.

\noindent In general, it may also turn to be useful to send a dedicated description of the state to the destination. The destination utilizes the recovered state as side information at the receiver. The coding scheme that we employ to establish the lower bound in Theorem~\ref{Theorem__LowerBound1__DiscreteChannel} uses block Markov encoding. In each block $i$, in addition to its message, the source also sends a two-layer description of the state $\dv s[i+2]$ to the relay and destination; a description $\hat{\dv s}_R[\iota_{Ri}]$ of $\dv s[i+2]$ intended to be recovered at the relay and a description $\hat{\dv s}_D[\iota_{Di}]$ of $\dv s[i+2]$ intended to be recovered at the destination. The two layers are possibly correlated. (In Remark~\ref{remark2}, we will comment on the delay of two blocks needed here). The relay guesses the source's message $w_i$ and the individual state description $\hat{\dv s}_R[\iota_{Ri}]$ dedicated to it from the source transmission and the previously recovered state description $\hat{\dv s}_R[\iota_{Ri-2}]$. It will then utilize the new state estimate as non-causal state at the encoder for collaborative source-relay binning in block $i+2$, through a combined decode-and-forward and Gelf'and-Pinsker binning. The destination guesses the source's message $w_{i-1}$ sent cooperatively by the source and relay and the individual state description $\hat{\dv s}_D[\iota_{Di-1}]$ which is dedicated to it from its output $(\dv y_3[i-1],\dv y_3[i])$ and the previously recovered state $(\hat{\dv s}_D[\iota_{Di-3}],\hat{\dv s}_D[\iota_{Di-2}])$, using a window-decoding method.
\end{remark}

\begin{remark}\label{remark2}
The source sends the descriptions intended to the relay and destination \textit{two blocks} ahead of time. That is, at the beginning of block $i$ the source describes the state vector $\dv s[i+2]$ to the relay and destination. While one block delay is sufficient to describe the state to the relay, a minimum of two blocks is necessary for the state reconstruction at the destination because of the used window decoding technique.
\end{remark}

We can generalize Theorem~\ref{Theorem__LowerBound1__DiscreteChannel} by allowing the source to also send a common description of the state which is intended to be recovered at both the relay and the destination (see \cite{ZS10}).

The following theorem provides a lower bound on the capacity of the state-dependent general discrete memoryless RC with informed source.

\vspace{0.1cm}

\begin{theorem}\label{Theorem__LowerBound2__DiscreteChannel}
The capacity of the discrete memoryless state-dependent relay channel with informed source is lower bounded by
\begin{align}
R^{\text{lo}} = \max \: \min\: \{&I(U,U_R;Y_3)-I(U,U_R;S),\nonumber\\
&I(U,U_R;Y_2,\hat{X})-I(U,U_R;S)-I(X;\hat{X})\}
\label{AchievableRate__Theorem__LowerBound2__DiscreteChannel}
\end{align}
subject to the constraint
\begin{align}
I(X;\hat{X}) &< I(U_R;Y_2,\hat{X}|U)-I(U_R;S|U)+[ I(U;Y_2,\hat{X})-I(U;S)]_{-}
\label{Constraint__AchievableRate__Theorem__LowerBound2__DiscreteChannel}
\end{align}
where $[x]_{-}=\min(x,0)$, and the maximization is over all joint measures on ${\mc S}\times{\mc U}\times{\mc U_R}\times{\mc X}_1\times{\mc X}_2\times{\mc X}\times\hat{\mc X}\times{\mc Y}_2\times{\mc Y}_3$ of the form
\begin{align}
&P_{S,U,U_R,X_1,X_2,X,\hat{X},Y_2,Y_3}\nonumber\\
&\quad = Q_SP_{U|S}P_{U_R|U,S}P_{X_1|U_R,U,S}P_{X|U,S}P_{\hat{X}|X}\mathds{1}_{X_2=\hat{X}}W_{Y_2,Y_3|X_1,X_2,S}.
\label{Measure__AchievableRate__Theorem__LowerBound2__DiscreteChannel}
\end{align}
\end{theorem}
\vspace{0.3cm}

\begin{remark}\label{remark5}
The rationale for the coding scheme which we use to obtain the lower bound in Theorem~\ref{Theorem__LowerBound2__DiscreteChannel} is as follows. With DF relaying, had the relay known the state then in each block the relay generates its input using the source transmission in the previous block and the state that controls the channel in the current block, as in \cite{KSS04}. For our model, the source knows what cooperative information, i.e., part of the message, the relay would send in each block. It also knows the state sequence that corrupts the transmission in that block. It can then generate the appropriate relay input vector that the relay would send had the relay known the state. The source can send this vector to the relay \textit{ahead of time}, and if the relay can estimate it to high accuracy, then collaborative source-relay binning in the sense of \cite{KSS04} is readily realized for transmission from the source and relay to the destination.
\end{remark}

\textbf{Outline of Proof:} A block Markov encoding with $B+1$ blocks is used.  Let us denote by $\dv x[k]$ the relay input carrying message $w_k \in [1,2^{nR}]$ that the relay would send in block $k$ had the relay known the state $\dv s[k]$, assuming DF relaying, with $k=2,\hdots,B+1$. Let us now consider transmission in two adjacent blocks $i$ and $i+1$. In the beginning of block $i$, the source sends information message $w_i$ of the current block, and, in addition, describes to the relay the input $\dv x[i+1]$ that the relay would send in the next block $i+1$ had the relay known the state $\dv s[i+1]$. Let $\hat{\dv x}[m_i]$ be a description of $\dv x[i+1]$. The source generates its input $\dv x_1[i]$ using two auxiliary codewords that are superimposed, a codeword $\dv u_R[i]$ that carries the index $m_i$ on top of a codeword $\dv u[i]$ that carries message $w_i$. Both codewords are selected using binning against the state $\dv s[i]$ that controls transmission in the current block $i$. The vector $\dv x[i+1]$, however, is the input that the relay would send in the next block $i+1$ had the relay known the state $\dv s[i+1]$, and so is generated at the source using binning against the state $\dv s[i+1]$. The description of vector $\dv x[i+1]$, which is sent to the relay in block $i$, is intended to combine coherently with the source transmission in block $i+1$. In the beginning of block $i$, the relay knows $m_{i-1}$ from the source transmission in previous block $i-1$, and sends $\dv x_2[i]=\hat{\dv x}[m_{i-1}]$.

\iffalse
Decoding at the relay and destination is as follows. At the end of block $i$, the relay estimates $m_i$ by $\hat{m}_i$ if there is a unique pair $(\hat{\dv u}[i],\hat{\dv u}_R[i])$ such that $\hat{\dv u}[i]$, $\hat{\dv u}_R[i]$ are jointly typical with $(\dv y_2[i],\hat{\dv x}[m_{i-1}])$, and $\hat{\dv u}_R[i]$ has $\hat{m}_i$ as message index. The destination waits until transmission in block $B+1$ is completed and then utilizes backward decoding to decode all messages, starting by message $w_{B+1}$. For block $b$, the destination decodes message $w_b$ not only from the codewords $\dv u[b]$ and $\hat{\dv x}[m_{b-1}]$, but also from the codeword $\dv u_R[b]$ that carries the description $\hat{\dv x}[m_b]$.
\fi

\begin{remark}\label{remark6}
In the scheme we described briefly in Remark~\ref{remark5}, the relay needs only estimate the code vector $\dv x[i]$ sent by the source in block $i-1$, and transmit the obtained estimate in the next block $i$. For instance, the relay does not need know the state sequence that actually controls the channel. Thus, transmission from the source terminal to the relay terminal can be regarded as that of an analog source which, in block $i$, produces a sequence $\dv x[i+1]$. This source has to be transmitted by the source terminal over a state-dependent channel and reconstructed at the relay terminal. The reconstruction error at the relay terminal influences the rate at which information can be decoded reliably at the destination by acting as an additional noise term.

\end{remark}

\iffalse
\begin{remark}\label{remark7}
A block Markov encoding is used to establish Theorem~\ref{Theorem__LowerBound2__DiscreteChannel}. In block $i$, the source transmits the message information $w_i$ and the index $m_i$ of a description $\hat{\dv x}[m_i]$ of the input $\dv x[i+1]$. In Theorem~\ref{Theorem__LowerBound2__DiscreteChannel}, the auxiliary random variable $U$ represents the message information and the auxiliary random variable $U_R$ represents the description information. In the proof in Appendix~\ref{appendixTheorem__LowerBound2__DiscreteChannel}, we generate the $U_R$'s codebook \textit{on top} of the $U$'s codebook. This may seem somewhat restrictive as the relay in general needs not decode the information message sent by the source as we indicated previously. However, this should not be regarded as such since our coding scheme as a whole is fashioned to mimic the functioning of a decode-and-forward relaying strategy in the symmetric state case, which itself performs well only when the channel to the relay is "better" than the channel to the destination.
\end{remark}
\fi

\section{The Gaussian RC with Informed Source}\label{secIV}

In this section, we consider a full-duplex state-dependent RC informed source in which the channel states and the noise are additive and Gaussian. In this model, the channel state can model an additive Gaussian interference which is assumed to be known (non-causally) to only the source. The channel outputs $Y_{2,i}$ and $Y_{3,i}$ at time instant $i$ for the relay and the destination, respectively, are related to the channel input $X_{1,i}$ from the source and $X_{2,i}$ from the relay, and the channel state $S_i$, by

\vspace{-0.4cm}

\begin{subequations}
\begin{align}
Y_{2,i}&=X_{1,i}+S_i+Z_{2,i}\\
Y_{3,i}&=X_{1,i}+X_{2,i}+S_i+Z_{3,i}.
\end{align}
\label{ChannelModel__StateDependent__GaussianRC}
\end{subequations}
The channel state $S_i$ is zero mean Gaussian random variable with variance $Q$; and only the source knows the state sequence $S^n$ (non-causally). The noises $Z_{2,i}$ and $Z_{3,i}$ are zero mean Gaussian random variables with variances $N_2$ and $N_3$, respectively; and are mutually independent and independent from the state sequence $S^n$ and the channel inputs $(X^n_1,X^n_2)$.

We shall also consider the following subclass of Gaussian RC with informed source, the parallel Gaussian RC with informed source and orthogonal components where $X_{1,i}=(X_{1R,i},X_{1D,i})$, $Y_{3,i}=(Y^{(1)}_{3,i},Y^{(2)}_{3,i})$ and
\begin{subequations}
\begin{align}
\label{OutputRelay__GaussianRC__OrthogonalComponents__Parallel}
Y_{2,i} &= X_{1R,i}+S_i+Z_{2,i}\\
\label{OutputDestination__Component1__GaussianRC__OrthogonalComponents__Parallel}
Y^{(1)}_{3,i} &= X_{1D,i}+S_i+Z^{(1)}_{3,i}\\
Y^{(2)}_{3,i} &= X_{2,i}+S_i+Z^{(2)}_{3,i},
\label{OutputDestination__Component2__GaussianRC__OrthogonalComponents__Parallel}
\end{align}
\label{ChannelModel__StateDependent__GaussianRC__OrthogonalComponents__Parallel}
\end{subequations}
where the noises $Z^{(1)}_{3,i}$ and $Z^{(2)}_{3,i}$ are zero mean Gaussian random variables with variances $N_3$, and are mutually independent and independent from the state sequence $S^n$ and the channel inputs $(X^n_1,X^n_2)$.

A parallel Gaussian RC with informed source and orthogonal components in which the state $S_i$ does not affect transmission from the relay to the destination will be said to be \textit{degenerate}. Its input-output relation is given by \eqref{ChannelModel__StateDependent__GaussianRC__OrthogonalComponents__Parallel} with \eqref{OutputDestination__Component2__GaussianRC__OrthogonalComponents__Parallel} substituted by $Y^{(2)}_{3,i} = X_{2,i}+Z^{(2)}_{3,i}$.

\iffalse
For the state-dependent Gaussian RC with informed source and orthogonal components, the channel outputs $Y_{2,i}$ and $Y_{3,i}$ at time instant $i$ for the relay and the destination, respectively, are related to the channel inputs $X_{1,i}=(X_{1R,i},X_{1D,i})$ and $X_{2,i}$ from the relay and the channel state $S_i$, by
\begin{subequations}
\begin{align}
Y_{2,i}&=X_{1R,i}+S_i+Z_{2,i}\\
Y_{3,i}&=X_{1D,i}+X_{2,i}+S_i+Z_{3,i}.
\end{align}
\label{ChannelModel__StateDependent__GaussianRC__OrthogonalComponents}
\end{subequations}
\fi

We consider the following individual power constraints on the average transmitted power at the source and the relay
\begin{equation}
\sum_{i=1}^{n}X_{1,i}^2 \leq nP_1, \qquad \sum_{i=1}^{n}X_{2,i}^2 \leq nP_2.
\label{IndividualPowerConstraintsFullDuplexRegime}
\end{equation}

\subsection{Upper Bound on the Capacity}\label{secIV_subsecB}

\begin{theorem}\label{Theorem__UpperBound__GeneralGaussianChannel}
The capacity of the state-dependent general Gaussian RC with informed source is upper-bounded by\begin{align}
&R^{\text{up}}_{\text{G}} = \max \min \Bigg\{\frac{1}{2}\log\Big(1+P_1(1-\rho^2_{12})(\frac{1}{N_2}+\frac{1}{N_3})\Big),\nonumber\\
&\frac{1}{2}\log\Big(1+\frac{(\sqrt{P_2}+\rho_{12}\sqrt{P_1})^2}{P_1(1-\rho^2_{12}-\varrho^2_{1s})+(\sqrt{\Delta_Q}+\varrho_{1s}\sqrt{P_1})^2+N_3}\Big)\nonumber\\
&+\frac{1}{2}\log\Big(1+\frac{P_1(1-\rho^2_{12}-\varrho^2_{1s})}{N_3}\Big)\Bigg\},
\label{UpperBound__GeneralGaussianChannel}
\end{align}
where $\Delta_Q=QN_2/((\sqrt{Q}+\sqrt{P_1})^2+N_2)$ and the maximization is over parameters $\rho_{12} \in [0,1]$, $\varrho_{1s} \in [-1,0]$ such that
\begin{equation}
\rho^2_{12}+\varrho^2_{1s} \leq 1.
\label{MaximizationRange__UpperBound__GeneralGaussianChannel}
\end{equation}
\end{theorem}
\vspace{0.3cm}

\noindent \textbf{Outline of Proof:} We only sketch the important steps, due to lack of space. The proof of the bound given by the first term of the minimization in \eqref{UpperBound__GeneralGaussianChannel} trivially follows by revealing the state $S^n$ to the relay and the destination. The proof of the bound given by the second term of the minimization in \eqref{UpperBound__GeneralGaussianChannel} is as follows. First, we show that there is an inevitable residual uncertainty at the relay about the state sequence $S^n$ after observing the channel outputs $Y_2^{i-1}=(Y_{2,1},\hdots,Y_{2,i-1})$. Then, considering transmission from the source and relay to the destination, we upper bound the sum rate that can be conveyed to the destination on the multiaccess part of the channel by accounting for the rate penalty that is caused by not knowing the state fully at the relay. In doing so, we assume that the message is revealed to the relay by a genie.

\iffalse
\begin{remark}\label{remark9}
In the upper bound in Theorem~\ref{Theorem__UpperBound__GeneralGaussianChannel}, for given $\rho_{12}$, the maximization over $\varrho_{1s}$ can in fact be restricted to either $\varrho_{1s}=-\sqrt{1-\rho^2_{12}}$, $\varrho_{1s}=0$, or a real root $\varrho_{1s}$ of the third-order polynomial $F(\varrho_{1s})$ that satisfies $\varrho_{1s} \in [-\sqrt{1-\rho^2_{12}},0]$, where
\begin{equation}
F(\varrho_{1s}) = A\varrho^3_{1s}+B\varrho^2_{1s}+C\varrho_{1s}+D
\label{Polynomial__maximization__over__rho1s__upperbound}
\end{equation}
with
\begin{align*}
A &= 4(P_1\Delta_Q)^{\frac{3}{2}}\\
B &= P_1\Delta_Q\Big[(\sqrt{P_2}+\rho_{12}\sqrt{P_1})^2+4(P_1(1-\rho^2_{12})+\Delta_Q+N_3)\Big]\\
C &= (P_1\Delta_Q)^{\frac{1}{2}}\Big[(\sqrt{P_2}+\rho_{12}\sqrt{P_1})^2+P_1(1-\rho^2_{12})+\Delta_Q+N_3\Big]\nonumber\\
&\:\:\:\:\:\:{\cdot}\Big[P_1(1-\rho^2_{12})+\Delta_Q+N_3\Big]\\
D &= \Delta_Q\Big[P_1(1-\rho^2_{12})+N_3\Big]\Big[\sqrt{P_2}+\rho_{12}\sqrt{P_1}\Big]^2.
\end{align*}
\end{remark}
\fi

\begin{remark}\label{remark10}
The established upper bound improves upon the cut-set upper bound through the second term of the minimization. The second term of the minimization is strictly tighter than that of the cut-set upper bound because it accounts for not knowing a part $\Delta_S^n$ of the state of power $\Delta_Q$ at the relay.
\end{remark}

\iffalse
\begin{corollary}\label{Corollary__UpperBound__DegradedGaussianChannel}
The capacity of the state-dependent degraded Gaussian RC with informed source is upper-bounded by
\begin{align}
&R^{\text{up}}_{\text{DG}} = \max \min \Bigg\{\frac{1}{2}\log\Big(1+\frac{P_1(1-\rho^2_{12})}{N_2}\Big),\nonumber\\
&\frac{1}{2}\log\Big(1+\frac{(\sqrt{P_2}+\rho_{12}\sqrt{P_1})^2}{P_1(1-\rho^2_{12}-\varrho^2_{1s})+(\sqrt{\Delta_Q}+\varrho_{1s}\sqrt{P_1})^2+N_3}\Big)\nonumber\\
&+\frac{1}{2}\log\Big(1+\frac{P_1(1-\rho^2_{12}-\varrho^2_{1s})}{N_3}\Big)\Bigg\},
\label{UpperBound__DegradedGaussianChannel}
\end{align}
where $\Delta_Q=QN_2/((\sqrt{Q}+\sqrt{P_1})^2+N_2)$ and the maximization is over parameters $\rho_{12} \in [0,1]$, $\varrho_{1s} \in [-1,0]$ such that
\begin{equation}
\rho^2_{12}+\varrho^2_{1s} \leq 1.
\end{equation}
\end{corollary}
\fi

\vspace{-0.1cm}
\subsection{Lower Bounds on the Capacity}\label{secIV_subsecC}
\vspace{-0.0cm}

\begin{theorem}\label{Theorem__LowerBound2__GeneralGaussianChannel}
The capacity of the state-dependent Gaussian RC with informed source is lower-bounded by
\begin{align}
R^{\text{lo}}_{\text{G}} = \max \frac{1}{2}\log\Big(1+\frac{(\sqrt{\bar{\gamma}P_1}+\sqrt{P_2-D})^2}{N_3+D+{\gamma}P_1}\Big),
\label{LowerBound2__GeneralGaussianChannel}
\end{align}
where
\begin{align}
D &:= P_2\frac{N_2}{N_2+{\gamma}P_1}
\label{Distortion__LowerBound2__GeneralGaussianChannel}
\end{align}
and the maximization is over $\gamma \in [0,1]$, with $\bar\gamma:=1-\gamma$.\\
\end{theorem}

The proof of Theorem~\ref{Theorem__LowerBound2__GeneralGaussianChannel} follows by using a coding scheme in which the source sends to the relay in block $i$ a quantized version of the input the relay would send in block $i+1$ had the relay known the state of the channel in that block, in the spirit of Theorem~\ref{Theorem__LowerBound2__DiscreteChannel}.

\iffalse
\textbf{Outline of Proof:}
The encoding and transmission scheme is as follows. For $0 \leq \gamma \leq 1$, let $X \sim \mc N(0,P_2)$ and $X_{1R} \sim \mc N(0,{\gamma}P_1)$, with $X$ jointly Gaussian with $S$ with $\mathbb{E}[XS]=0$; and $X_{1R}$ jointly Gaussian with $(S,X)$, with $\mathbb{E}[X_{1R}S]=\mathbb{E}[X_{1R}X]=0$. Also, let $0 \leq D \leq P_2$ be given, and consider the test channel $\hat{X}=aX+\tilde{X}$, where  $a:=1-D/P_2$ and $\tilde{X}$ is a Gaussian random variable with zero mean and variance $\tilde{P}_2=D(1-D/P_2)$, independent from $X$ and $S$. Using this test channel, we calculate $\mathbb{E}[(X-\hat{X})^2]=D$ and $\mathbb{E}[\hat{X}^2]=P_2-D$.

\noindent We use the following two random variables to generate the auxiliary codewords $U_i$ and $U_{R,i}$ which we will use in the sequel.
\begin{align}
\label{JointSourceRelay__DPC__LowerBound2}
U &= \Big(\sqrt{\frac{\bar{\gamma}P_1}{P_2}}+\sqrt{\frac{P_2-D}{P_2}}\Big)X+{\alpha}S\\
U_R &= X_{1R}+\alpha_R\Big(S+\frac{\sqrt{\bar{\gamma}P_1}}{\sqrt{\bar{\gamma}P_1}+\sqrt{P_2-D}}X\Big),
\label{JointSourceRelay__DPC__LowerBound2}
\end{align}
 where
\begin{align}
\alpha =& \frac{(\sqrt{\bar{\gamma}P_1}+\sqrt{P_2-D})^2}{(\sqrt{\bar{\gamma}P_1}+\sqrt{P_2-D})^2+(N_3+D+{\gamma}P_1)}\\
\alpha_R &=\frac{{\gamma}P_1}{{\gamma}P_1+N_2}.
\end{align}
\noindent A block Markov encoding is used. For each block $i$, let $\dv x[i]$ be a Gaussian signal which carries message $w_i \in [1,2^{nR}]$ and is obtained via a DPC considering $\dv s[i]$ as noncausal channel state information, as
\begin{equation}
\Big(\sqrt{\frac{\bar{\gamma}P_1}{P_2}}+\sqrt{\frac{P_2-D}{P_2}}\Big) \dv x[i] = \dv u[i] - {\alpha}\dv s[i],
\end{equation}
where the components of $\dv u[i]$ are generated i.i.d. using the auxiliary random variable $U$.
For every block $i$, the source quantizes $\dv x[w_i]$ into $\hat{\dv x}[m_i]$, where $m_i \in [1,2^{n\hat{R}}]$. Using the above test channel, the source can encode $\dv x[w_i]$ successfully at the quantization rate
\begin{align}
\hat{R} &=I(X;\hat{X})\nonumber\\
        &= \frac{1}{2}\log(\frac{P_2}{D}).
\label{QuantizationRate__LowerBound2__GeneralGaussianChannel__SourceCodingConstraint}
\end{align}

Let $m_i$ be the index associated with $\dv x[w_{i+1}]$. In the beginning of block $i$, the source sends a superposition of two Gaussian vectors,
\begin{equation}
\dv x_1[i]=\dv x_{1R}[m_i]+\sqrt{\frac{\bar{\gamma}P_1}{P_2}}\dv x[w_i].
\label{SourceInput__LowerBound2__GeneralGaussianChannel}
\end{equation}
In equation \eqref{SourceInput__LowerBound2__GeneralGaussianChannel}, the signal $\dv x_{1R}[m_i]$ carries message $m_{i}$ and is obtained via a DPC considering $(s[i],\dv x[w_i])$ as noncausal channel state information, as
\begin{align}
\dv x_{1R}[m_i] &=\dv u_R[i]-\alpha_R\Big(\dv s[i]+\sqrt{\frac{\bar{\gamma}P_1}{P_2}}\dv x[w_i]\Big),\label{JointSourceRelay__DPC__LowerBound2}
\end{align}
where the components of $\dv u_R[i]$ are generated i.i.d. using the auxiliary random variable $U_R$.

\noindent In the beginning of block $i$, the relay has decoded message $m_{i-1}$ correctly (this will be justified below) and sends
\begin{equation}
\dv x_2[i]=\frac{\sqrt{P_2}}{\sqrt{P_2-D}}\hat{\dv x}[m_{i-1}].
\end{equation}

For the decoding arguments at the source and the relay, we give simple arguments based on intuition (the rigorous decoding uses joint typicality). Also, since all the random variables are i.i.d., we sometimes omit the time index. The relay decodes the index $m_i$ from the received $\dv y_2[i]$ at the end of block $i$. Since signal $\dv x_{1R}[m_i]$ is precoded at the source against the interference caused by the information message $w_i$, decoding at the relay can be done reliably as long as $n$ is large and
\begin{align}
\hat{R} \leq \frac{1}{2}\log\Big(1+\frac{{\gamma}P_1}{N_2}\Big).
\label{QuantizationRate__LowerBound2__GeneralGaussianChannel__ChannelCodingConstraint}
\end{align}
The destination decodes message $w_i$ from the received $\dv y_3[i]$ at the end of block $i$, considering signal $\dv x_{1R}[m_i]$ as unknown noise, with
\begin{align}
\dv y_3[i] &= \dv x_1[i] + \dv x_2[i] + \dv s[i] +\dv z_3[i]\nonumber\\
&= \Big(\sqrt{\frac{\bar{\gamma}P_1}{P_2}}\dv x[w_i]+\sqrt{\frac{P_2}{P_2-D}}\hat{\dv x}[m_{i-1}]\Big)+\dv s[i]+(\dv z_3[i]+\dv x_{1R}[m_i]).
\label{OutputDestination__LowerBound2__GeneralGaussianChannel}
\end{align}
Let now ${\dv x}'[i]$ be the optimal linear estimator of $\Big(\sqrt{\frac{\bar{\gamma}P_1}{P_2}}\dv x[w_i]+\sqrt{\frac{P_2}{P_2-D}}\hat{\dv x}[m_{i-1}]\Big)$ given $\dv x[w_i]$ under minimum mean square error criterion, and $\dv e_{\dv x}[i]$ the resulting estimation error. The estimator $\hat\hat{\dv x}[i]$ and the estimation error $\dv e_{\dv x}[i]$ are given by
\begin{align}
{\dv x}'[i] &= \mathbb{E}\Big[\sqrt{\frac{\bar{\gamma}P_1}{P_2}}\dv x[w_i]+\sqrt{\frac{P_2}{P_2-D}}\hat{\dv x}[m_{i-1}]|\dv x[i]\Big]\nonumber\\
            &= \Big(\sqrt{\frac{\bar{\gamma}P_1}{P_2}}+\sqrt{\frac{P_2-D}{P_2}}\Big)\dv x[w_i]\\\dv e_{\dv x}[i] &= \sqrt{\frac{P_2}{P_2-D}}\hat{\dv x}[m_{i-1}]-\sqrt{\frac{P_2-D}{P_2}}\dv x[w_i].
\end{align}

\noindent We can alternatively write the output $\dv y_3[i]$ in \eqref{OutputDestination__LowerBound2__GeneralGaussianChannel} as
\begin{align}
\dv y_3[i] = \xi\dv x[w_i]+\dv s[i]+\Big(\dv z_3[i]+\dv e_{\dv x}[i]+\dv x_{1R}[m_i]\Big),
\label{EquivalentOutputDestination__LowerBound2__GeneralGaussianChannel}
\end{align}
where
\begin{equation}
\xi := \sqrt{\frac{\bar{\gamma}P_1}{P_2}}+\sqrt{\frac{P_2-D}{P_2}}
\end{equation}
and $\dv e_{\dv x}[i]$ is Gaussian with variance $D$ and is independent of $\dv x[w_i]$ and $\dv s[i]$.

\noindent Now, considering the equivalent form \eqref{EquivalentOutputDestination__LowerBound2__GeneralGaussianChannel} of the output $\dv y_3[i]$, it is easy to see that the destination can decode message $w_i$ correctly at the end of block $i$ as long as $n$ is large and
\begin{align}
R &\leq I(U;Y_3)-I(U;S)\nonumber\\
  &= \frac{1}{2}\log\Big(1+\frac{(\sqrt{\bar{\gamma}P_1}+\sqrt{P_2-D})^2}{N_3+D+{\gamma}P_1}\Big).
\label{Decoding__at__Destination__LowerBound2__GeneralGaussianChannel}
\end{align}

\noindent Furthermore, combining \eqref{QuantizationRate__LowerBound2__GeneralGaussianChannel__SourceCodingConstraint} and \eqref{QuantizationRate__LowerBound2__GeneralGaussianChannel__ChannelCodingConstraint} we get
\begin{equation}
D \geq P_2\frac{N_2}{N_2+{\gamma}P_1}.
\label{MinimumDistortion__LowerBound2__GeneralGaussianChannel}
\end{equation}
Finally, observing that the RHS of \eqref{Decoding__at__Destination__LowerBound2__GeneralGaussianChannel} decreases with $D$, we obtain \eqref{LowerBound2__GeneralGaussianChannel} by taking the equality in \eqref{MinimumDistortion__LowerBound2__GeneralGaussianChannel} and maximizing the RHS of \eqref{Decoding__at__Destination__LowerBound2__GeneralGaussianChannel} over $\gamma \in [0,1]$. This completes the proof. \begin{flushright} \hfill \ensuremath{\Box} \end{flushright}
\fi

\iffalse
\begin{remark}\label{remark5}
The rationale for the coding scheme which we use to obtain the lower bound in Theorem~\ref{Theorem__LowerBound2__GeneralGaussianChannel} is as follows. With DF relaying, had the relay known the state then in each block the relay generates its input using the source transmission in the previous block and the state that controls the channel in the current block, as in \cite{KSS04}. For our model, the source knows what cooperative information, i.e., part of the message, the relay should send in each block. It also knows the state sequence that corrupts the transmission in that block. It can then generate the appropriate relay input vector that the relay would send had the relay known the state. The source can send this vector to the relay \textit{ahead of time}, and if the relay can estimate it to high accuracy, then collaborative source-relay binning in the sense of \cite{KSS04} is readily realized for transmission from the source and relay to the destination.
\end{remark}

\textbf{Outline of Proof:} A block Markov encoding with $B+1$ blocks is used. Let us denote by $\dv x[w_i]$ the i.i.d Gaussian signal (with power $P_2$) carrying message $w_i \in [1,2^{nR}]$ that the relay would send in block $k$ had the relay known the state $\dv s[k]$, assuming DF relaying, with $k=2,\hdots,B+1$. Let us now consider transmission in two adjacent blocks $i$ and $i+1$. In the beginning of block $i$, the source sends the message $w_i$ of the current block (with a fraction $\bar{\gamma}P_1$ of its power), and, in addition, describes to the relay the input $\dv x[w_{i+1}]$ that the relay would send in the next block $i+1$ had the relay known the state $\dv s[i+1]$ (with the remaining ${\gamma}P_1$ of its power). Let $\hat{\dv x}[m_i]$ be a description of $\dv x[w_{i+1}]$, obtained with the test channel $\hat{X}=aX+\tilde{X}$, where  $a:=1-D/P_2$ and $\tilde{X}$ is a Gaussian random variable with zero mean and variance $\tilde{P}_2=D(1-D/P_2)$, independent from $X$ and $S$. The input from the source in the beginning of block $i$ is $\dv x_1[i]=\dv x_{1R}[m_i]+\sqrt{\bar{\gamma}P_1/P_2}\dv x[w_i]$; where $\dv x[w_i]$ is precoded using binning against the state $\dv s[i]$ that controls transmission in the current block $i$, and $\dv x_{1R}[m_i]$ carries the index $m_i$ and is is precoded against both the state $\dv s[i]$ and the information message interference $\sqrt{\bar{\gamma}P_1/P_2}\dv x[w_i]$. In the beginning of block $i$, the relay knows $m_{i-1}$ from the source transmission in previous block $i-1$, and $\dv x_2[i]=\sqrt{P_2}/\sqrt{P_2-D}\hat{\dv x}[m_{i-1}]$. The vector $\dv x[w_{i+1}]$ is the input that the relay would send in the next block $i+1$ had the relay known the state $\dv s[i+1]$, and so is generated at the source using binning against the state $\dv s[i+1]$. The vector $\dv x[w_{i+1}]$, and its description $\hat{\dv x}[m_i]$ which is sent to the relay in block $i$ is intended to combine coherently with the source transmission in block $i+1$. The destination decodes blockwise. At the end of block $i$, the destination decodes message $w_i$ from the cooperative transmission, considering the description-bearing signal $\dv x_{1R}[m_i]$ as unknown noise.

\begin{remark}\label{remark6}
In the scheme we briefly outlined, the relay needs only estimate the code vector $\dv x[w_i]$ described by the source in block $i-1$, and transmit the obtained estimate in the next block $i$. For instance, the relay does not need know the cooperative information that the estimated vector actually carries, let alone the state sequence that controls the channel. Thus, transmission from the source terminal to the relay terminal can be regarded as that of an analog source which, in block $i$, produces a sequence $\dv x[w_{i+1}]$. This source has to be transmitted by the source terminal over a state-dependent channel and reconstructed at the relay terminal. The reconstruction error at the relay terminal influences the rate at which information can be decoded reliably at the destination by acting as an additional noise term.

\end{remark}
\fi

\iffalse
\begin{remark}\label{remark7}
A block Markov encoding is used to establish Theorem~\ref{Theorem__LowerBound2__DiscreteChannel}. In block $i$, the source transmits the message information $w_i$ and the index $m_i$ of a description $\hat{\dv x}[m_i]$ of the input $\dv x[i+1]$. In Theorem~\ref{Theorem__LowerBound2__DiscreteChannel}, the auxiliary random variable $U$ represents the message information and the auxiliary random variable $U_R$ represents the description information. In the proof in Appendix~\ref{appendixTheorem__LowerBound2__DiscreteChannel}, we generate the $U_R$'s codebook \textit{on top} of the $U$'s codebook. This may seem somewhat restrictive as the relay in general needs not decode the information message sent by the source as we indicated previously. However, this should not be regarded as such since our coding scheme as a whole is fashioned to mimic the functioning of a decode-and-forward relaying strategy in the symmetric state case, which itself performs well only when the channel to the relay is "better" than the channel to the destination.
\end{remark}
\fi

\begin{remark}\label{remar11}
It is insightful to observe that the rate in Theorem~\ref{Theorem__LowerBound2__GeneralGaussianChannel} does not depend on the strength of the state $S$. This makes the described coding scheme appreciable, particularly for the case of arbitrary strong interference for which classical coding schemes have the relay sending at no positive rate when it operates in decode-and-forward mode, because of the unknown interference.
\end{remark}

We now turn to establish a lower bound on the capacity of the state-dependent Gaussian RC using the idea of state description.

\begin{definition}\label{definition1} Let
\begin{align}
\tilde{Q}_S(t,Q,D) &:=(1-t)^2Q-t(t-2)D\nonumber\\
R(\alpha,P,Q,N) &:= \frac{1}{2}\log\Big(\frac{P(P+Q+N)}{PQ(1-\alpha)^2+N(P+\alpha^2Q)}\Big)\nonumber
\end{align}
for non-negative $t, D, P, Q, N$, and $\alpha \in \mc A(P,Q,N):=\{x \in \mathbb{R}\::\: R(x,P,Q,N) \geq 0\}$.
\end{definition}

\vspace{0.2cm}

\begin{theorem}\label{Theorem__LowerBound1__GeneralGaussianChannel}
The capacity of the state-dependent Gaussian RC with informed source is lower-bounded by
\begin{align}
&R^{\text{lo}}_{\text{G}} = \max\:\min\:\big\{R(\alpha,\beta\bar{\gamma}P_1,\tilde{Q},N_2+{\gamma}P_1), \nonumber\\
&R(\alpha,\beta\bar{\gamma}P_1,\tilde{Q},N_3+{\gamma}P_1)+\frac{1}{2}\log\Big(1+\frac{(\sqrt{\bar\beta\bar{\gamma}P_1}+\sqrt{P_2})^2}{N_3+D+{\gamma}P_1+\beta\bar{\gamma}P_1}\Big)\big\},
\label{LowerBound1__GeneralGaussianChannel}
\end{align}
where
\begin{align}
\label{Distortion__LowerBound1__GeneralGaussianChannel}
D &= Q\frac{N_2}{N_2+{\gamma}P_1}\\
\tilde{Q} &= \tilde{Q}_S(\alpha_2,Q,D)\\
\alpha_2 =& \frac{(\sqrt{\bar\beta\bar{\gamma}P_1}+\sqrt{P_2})^2}{(\sqrt{\bar\beta\bar{\gamma}P_1}+\sqrt{P_2})^2+\beta\bar{\gamma}P_1+(N_3+D+{\gamma}P_1)}
\end{align}
and the maximization is over $\beta \in [0,1]$, $\gamma \in [0,1]$ and  $\alpha \in \mc A(\beta\bar{\gamma}P_1,\tilde{Q},N_2+{\gamma}P_1) \cap \mc A(\beta\bar{\gamma}P_1,\tilde{Q},N_3+{\gamma}P_1)$.
\end{theorem}

\vspace{0.2cm}

\textbf{Outline of Proof of Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel}:} An outline of proof of Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel} is as follows. The result in Theorem~\ref{Theorem__LowerBound1__DiscreteChannel} for the DM case can be extended to memoryless channels with discrete time and continuous alphabets using standard techniques \cite[Chapter 7]{G68}. For the state-dependent Gaussian relay channel \eqref{ChannelModel__StateDependent__GaussianRC}, we evaluate the rate \eqref{AchievableRate__Theorem__LowerBound1__DiscreteChannel} with the following choice of input distribution. We choose $\hat{S}_D=\O$, $U_D=\O$. Furthermore, we consider the test channel $\hat{S}_R=aS+\tilde{S}_R$, where  $a:=1-D/Q$ and $\tilde{S}_R$ is a Gaussian random variable with zero mean and variance $\sigma^2_{\tilde{S}_R}=D(1-D/Q)$, independent from $S$. The random variable $X_2$ is Gaussian with zero mean and variance $P_2$, independent of $S$ and $\hat{S}_R$. The random variable $X_1$ is composed of two parts, $X_1=X_{1R}+X$, where $X_{1R}$ is Gaussian with zero mean and variance ${\gamma}P_1$, for some $\gamma \in [0,1]$, is independent of $S$, $\hat{S}_R$, $X_2$; and $X=\sqrt{\bar\beta\bar{\gamma}P_1/P_2}X_2+X'$, where $X'$ is Gaussian with zero mean and variance $\beta\bar{\gamma}P_1$, for some $\beta \in [0,1]$, and is independent of $X_{1R}$, $X_2$ and $(S,\hat{S}_R)$. The auxiliary random variables are chosen as
\begin{subequations}
\begin{align}
V &= \Big(\sqrt{\frac{\bar\beta\bar{\gamma}P_1}{P_2}}+1\Big)X_2+\alpha_2\hat{S}_R \\
U &= X' +\alpha (S-\alpha_2\hat{S}_R)\\
U_R &= X_{1R}+\alpha_R(1-\alpha)S
\end{align}
\label{RandomVariables__LowerBound1__GeneralGaussianChannel}
\end{subequations}
where
\begin{subequations}
\begin{align}
\alpha_{2} &= \frac{(\sqrt{\bar\beta\bar{\gamma}P_1}+\sqrt{P_2})^2}{(\sqrt{\bar\beta\bar{\gamma}P_1}+\sqrt{P_2})^2+\beta\bar{\gamma}P_1+(N_3+D+{\gamma}P_1)}\\
\alpha_R &= \frac{{\gamma}P_1}{{\gamma}P_1+N_2}, \:\:\: D := Q\frac{N_2}{N_2+{\gamma}P_1}.
\end{align}
\end{subequations}

\iffalse
\begin{remark}\label{remark12}
The parameter $\alpha$ in Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel} stands for DPC's scale factor in precoding the information message against the interference on its way to the relay and to the destination. Because the model \eqref{ChannelModel__StateDependent__GaussianRC} has the links to the relay and to the destination corrupted  by noise terms with distinct variances, one cannot remove the effect of the interference on the two links simultaneously via one single DPC as in \cite{KELW07}. This explains why the parameter $\alpha$ is left to be optimized over in \eqref{LowerBound1__GeneralGaussianChannel}. However, in the spirit of \cite{KELW07}, one can improve the rate of Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel} by time sharing coding schemes that are similar to the one we employed for Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel} but with different inflation parameters tailored respectively for the link to the relay and the link to the destination, as in \cite{ZV09b}.
\end{remark}
\fi

\vspace{-0.1cm}
\subsection{Analysis of Some Special Cases}\label{secIV__subsecD}
\vspace{-0.0cm}

\begin{corollary}\label{Corollary__DegenerateParallel__GaussianRC__OrthogonalComponents}
The capacity of the degenerate parallel Gaussian RC with informed source and orthogonal components is given by

\vspace{-0.6cm}

\begin{align}
\mc C_{\text{G-DegParOrth}} = \max_{0 \leq \gamma \leq 1} \min \big\{&\frac{1}{2}\log(1+\frac{{\gamma}P_1}{N_2}),\frac{1}{2}\log(1+\frac{P_2}{N_3})\big\}\nonumber\\
&+\frac{1}{2}\log(1+\frac{{(1-\gamma)}P_1}{N_3}).
\label{Capacity__DegenerateParallel__GaussianRC__OrthogonalComponents}
\end{align}
\end{corollary}

\begin{itemize}
\item[1)] If $N_2 \longrightarrow 0$, the upper bound of Theorem~\ref{Theorem__UpperBound__GeneralGaussianChannel} and the lower bound of Theorem~\ref{Theorem__LowerBound2__GeneralGaussianChannel} tend asymptotically to the same value

\vspace{-0.5cm}

\begin{equation}
\mc C_{\text{G}} =\frac{1}{2}\log\Big(1+\frac{(\sqrt{P_1}+\sqrt{P_2})^2}{N_3}\Big)-o(1)
\label{LowerBound__ExtremeCase__CleanRelay}
\end{equation}
where $o(1) \longrightarrow 0$ as $N_2 \longrightarrow 0$.

\item[2)] If $N_2 \longrightarrow \infty$, the upper bound of Theorem~\ref{Theorem__UpperBound__GeneralGaussianChannel} tends to
\begin{align}
\mc C_{\text{G}} &= \frac{1}{2}\log(1+\frac{P_1}{N_3}),
\label{LowerBound__ExtremeCase__NoisyRelay}
\end{align}
which is achieved by standard DPC at the source.
\item[3)] \textit{Arbitrarily strong channel state:} In the asymptotic case $Q \rightarrow \infty$, the lower bound of Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel} tends to
\begin{equation}
R^{\text{lo}}_{\text{G}} =\frac{1}{2}\log\big(1+\frac{P_1}{\max(N_2,N_3)}\big).
\end{equation}
\end{itemize}

\vspace{-0.3cm}
\subsection{Numerical Examples and Discussion}\label{secIV_subsecD}
\vspace{-0.0cm}

\iffalse
\begin{figure}[!htpb]
\begin{center}
\includegraphics[width=\linewidth]{Fig1__General__RC.pdf}
\end{center}
\caption{Illustration of the upper bound in Theorem~\ref{Theorem__UpperBound__GeneralGaussianChannel} and the lower bound of Theorem~\ref{Theorem__LowerBound2__GeneralGaussianChannel} for the state-dependent General Gaussian RC with informed source \eqref{ChannelModel__StateDependent__GaussianRC} versus the SNR in the link source-to-relay. Numerical values are: $P_1=N_3=10$ dB, $P_2=5$ dB, $Q=30$ dB.}
\label{Fig1__IllustrativeExamples__General__RC}
\end{figure}
\fi

Figure~\ref{Fig2__IllustrativeExamples__General__RC} illustrates the upper bound of Theorem~\ref{Theorem__UpperBound__GeneralGaussianChannel}, the lower bound of Theorem~\ref{Theorem__LowerBound2__GeneralGaussianChannel} and the lower bound of Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel} for the model \eqref{ChannelModel__StateDependent__GaussianRC} as functions of $\text{SNR}=P_1/N_2$ (in decibels). Also shown for comparison are the cut-set upper bound and the trivial lower bound obtained by considering the channel state as unknown noise and implementing classic DF at the relay. The figure shows that the lower bound  \eqref{LowerBound2__GeneralGaussianChannel} of Theorem~\ref{Theorem__LowerBound2__GeneralGaussianChannel} is asymptotically optimal in $\text{SNR}$. Note that it outperforms the lower bound \eqref{LowerBound1__GeneralGaussianChannel} of Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel} for almost all $\text{SNR}$ values. Also, the upper bound \eqref{UpperBound__GeneralGaussianChannel} is strictly better than the cut-set upper bound, as we indicated in the proof of  Theorem~\ref{Theorem__UpperBound__GeneralGaussianChannel}.

\iffalse
The figure also shows that the upper bound \eqref{UpperBound__GeneralGaussianChannel} is strictly better than the cut-set upper bound. The improvement is due to that the upper bound \eqref{UpperBound__GeneralGaussianChannel} accounts for some inevitable residual uncertainty at the relay about the state after observing its output sequence, as we indicated in the proof of  Theorem~\ref{Theorem__UpperBound__GeneralGaussianChannel}.
\fi

\begin{figure}[!htpb]

        \begin{minipage}[t]{\linewidth}
        \vspace{-0.5cm}
        \begin{center}
        \subfigure[]
        {
	\includegraphics[width=\linewidth,height=0.55\linewidth]{Fig1__General__RC.pdf}
        \label{Fig2c__IllustrativeExamples__General__RC}
        }
        \subfigure[]
        {
        \vspace{-0.3cm}
        \includegraphics[width=\linewidth,height=0.55\linewidth]{Fig2b__General__RC.pdf}
        \label{Fig2b__IllustrativeExamples__General__RC}
        }
        \vspace{-0.3cm}
        \caption{Lower and upper bounds on the capacity of the state-dependent General Gaussian RC with informed source versus the SNR in the link source-to-relay, for two examples of numerical values (a) $P_1=N_3=10$ dB, $P_2=5$ dB, $Q=30$ dB, and (b) $P_2=40$ dB, $P_1=Q=N_3=10$ dB.}\vspace{-0.3cm}
        \label{Fig2__IllustrativeExamples__General__RC}
        \end{center}
        \vspace{-0.3cm}
        \end{minipage}
\vspace{-0.6cm}
\end{figure}

\iffalse
\begin{figure}[!htpb]
\begin{center}
        \includegraphics[width=\linewidth]{/hlocal/zaidi/Work/RelayChannel/InformedSourceOnly/WorkWithShlomo/Figs/Fig2b__General__RC.pdf}
\end{center}
        \caption{Lower and upper bounds on the capacity of the state-dependent General Gaussian RC with informed source versus the SNR in the link source-to-relay, for two examples of numerical values (a) $P_2=40$ dB, $P_1=Q=N_3=10$ dB, and (b) $P_1=5$ dB, $P_2=255$ dB, $Q=N_3=10$ dB.}
\label{Fig2__IllustrativeExamples__General__RC}
\end{figure}
\fi

\iffalse
\begin{remark}\label{remark13}
Recalling that the upper bound \eqref{UpperBound__GeneralGaussianChannel} improves upon the cut-set upper bound through the second term of the minimization (see Remark~\ref{remark10}), the fact that the improvement is small or is not visible at very small and very large values of $\text{SNR}$ can be explained as follows. The second term of the minimization in \eqref{UpperBound__GeneralGaussianChannel} accounts for the relay's not knowing a part of the state of power $\Delta_Q=QN_2/((\sqrt{Q}+\sqrt{P_1})^2+N_2)$. For very large values of $\text{SNR}$ (e.g., $N_2$ arbitrarily small), this unknown part of the state is negligible. For very small values of $\text{SNR}$ (e.g., $N_2$ arbitrarily large), the unknown part of the state at the relay is strong and the second term of the minimization in \eqref{UpperBound__GeneralGaussianChannel} is strictly tighter than that of the cut-set upper bound but, in this case, the upper bound \eqref{UpperBound__GeneralGaussianChannel} and the cut-set upper bound are both given by their first term of minimization which is common and which becomes operative at this range of SNR.
\end{remark}
\fi

\iffalse
\begin{remark}\label{remark14}
The lower bound of Theorem~\ref{Theorem__LowerBound2__GeneralGaussianChannel} is asymptotically close to optimal in $\text{SNR}$ as we mentioned in the "Extremes Cases Analysis" section and is visible from Figure~\ref{Fig2__IllustrativeExamples__General__RC}. This is because the appropriate relay input, which is precoded at the source against the state and is encoded in a manner that it should combine coherently with the source transmission in next block, can be sent by the source to the relay at almost no expense in power and can be learned by the relay with negligible distortion in this case. One can be tempted to expect a similar behavior for the lower bound of Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel} since, for the latter as well, the relay can learn a "good" estimate of the state at almost no expense in source's power and with negligible distortion. This should not however be since our coding scheme for Theorem~\ref{Theorem__LowerBound1__GeneralGaussianChannel} requires the relay to also decode the source's information message. Related to this aspect, the effect of the limitation which we mentioned in Remark~\ref{remark12} is visible at large $\text{SNR}$ for this lower bound.

\end{remark}
\fi

\vspace{-0.1cm}
\section*{Acknowledgement}
\vspace{-0.0cm}
This work has been supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless Communications. A. Zaidi and L. Vandendorpe also thank the concerted Action SCOOP for funding. The work of S. Shamai has also been supported by the CORNET consortium.

\vspace{-0.2cm}

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


Home > Instructional Materials / Teaching Aids > Social Science > Biographies Book - Our Big, Cool USA!
Fun! Contemporary! Colorful! Books and map are excellent products about Our, Big Cool USA, ALL 50 states, and ALL U.S. territories! Educational facts, activities, and visual aids make this series perfect for a supplement to U.S. History lessons or a stand-alone lesson. Makes U.S. studies fun, big, and cool!Biographies Book.


A Russian man wants a new sport to try. He wants to make a new way to do snowboarding. He tries to do it with vehicles. The vehicles cannot go where he wants to go.
He realises that horses can go in more places. He tries snowboarding with horses. He invents horse-boarding. This is when people tie themselves to horses. Then the horses drag them through the snow.
The horse-boarders do not want to make money from this sport yet. For now, they are just horsing about!
Difficult words: snowboard (a sport where you ride on a board through snow), realise (become aware), horse about (play).


text:
Q: Jenkins Workflow Optional Node We have a section of our pipeline that needs to be run on slaves running windows so we have a section similar to:
node("Windows") {
   // Do some windows stuff
}

Because this part of the pipeline isn't mission critical I'd prefer the whole job not to fail if the Windows slaves are down for some reason.
Is there some way of having some logic here that says "do this if a Windows slave is up but skip if there all down".

A: You can use Conditional Build Step plugin. So you can wrap you windows part of workflow script into slave validation of some sort.

A: The answer to this question gave me the information I needed.
Jenkins How to find if a given slave is running a Job
So I was able to use:
node.getComputer().isOffline()

redpajama_set_name: RedPajamaStackExchange


text:
Steven Evans, Sunil Gupta
INDIA: Contemporary Photographic and New Media Art
Steven Evans Sunil Gupta
This book presents 48 contemporary artists and collectives working in dialogue with the long history and emergent future of India and its people. Its focus lies on the contemporary moment through a range of approaches, including art photography, new media, installation, moving image, journalism, and documentary photography. Themes include caste and class, the partitioning of the subcontinent, gender and sexuality, activism and conflict, racism, religion, nationalism, new technologies and development, the environment, human settlement, migration, and integration. "As a large, multilingual subcontinent, India has always relied on images to maintain a cohesive whole across myriad subcultures, regions, castes and languages. The introduction of photomechanical imaging in the nineteenth century enabled the rapid reproduction and dissemination of both spiritual and scientific ideals. Photography for most of its history was too expensive and technical, and was left in the hands of 'experts'-until the birth of digital technologies. The book addresses the legacy of the last twenty years, a period when photography and moving image media have been consistently included within critical exhibitions of fine art." (Sunil Gupta, Lead Curator) INDIA: Contemporary Photographic and New Media Art is edited by Steven Evans, FotoFest Executive Director and curator, with Sunil Gupta, Delhi-born artist and project lead curator, and includes essays from experts on the region and contemporary art.
Steven EvansSunil Gupta
Schilt Publishing B.V.
BoekenEngelse boekenKunst & CultuurFotografie & DesignFotografie, film, videokunst

redpajama_set_name: RedPajamaCommonCrawl


Fresher's day was organized on 20 September 2017 by Aditya Bangalore Institute of Pharmacy Education and Research Bangalore. Fresher's day was arranged in a grand manner with games, personality contest, various cultural & other activities .The management, staff and senior students heartily welcomed the first year students. The programme helped the juniors familiarize them with the senior students.


Today we have an early start as we have to arrive at the departure terminal of our charter aircraft company to check our gear in and complete our loading manifests prior to our flight to Kokoda. Air charter companies are keen to depart Port Moresby by 8.00 AM as weather conditions are most favourable around this time.
The flight across the Owen Stanley Ranges is 96 km 'as the crow flies' and takes about 25 minutes to complete.
Our PNG support crew will be waiting for us at the Kokoda airfield. We then get a chance to meet with them and marry up with your personal carrier if you have booked one.
After we have sorted our gear we trek up to the Kokoda plateau and inspect the area where Bravo Company from the 39th Battalion was attacked on the night of 29 July 1942. The average age of one of the sections in Bravo Company was 18½ years.
Your Trek Leader will provide you with a detailed briefing on the battle that saw the Commanding Officer of the 39th Battalion killed in action as he led his young troops from the front.
A local Orokaiva dance group will provide a traditional welcome with kundu drums beating and shouts of 'Oro', 'Oro', 'Oro' which means 'Welcome', 'Welcome', Welcome'.
We then follow the footsteps of the young diggers as they made their way through the misty surrounds of the rubber plantation as the Japanese swarmed over their position.
We continue trekking through to Kovello village and onto our campsite at Hoi village where we have plenty of time to make any adjustments to our backpacks, re-sort our gear if necessary and get aqainted with organising our individual tents which will already have been erected by our PNG campsite crew.


I really enjoyed the first book in this series, so was pleased to find this one. I can't remember Sky from that book, but I certainly recall the cheating Bradley. Poor Sky, that's awful to find, that the person you love and thought loved you has been playing away…She did the right thing in giving him the elbow but she's still confused about her feelings. Of course that's natural, you can't just switch off years of love – it's not that easy.
Along with wedding plans that go astray there's the wonderful Hayden. I love the way he "drops" in – just when the group are telling Sky she won't get a man fall from the sky into her lap, Hayden falls in – through the ceiling! Fabulous. Hayden is perfect, he can see that Sky is cautious, and just gently keeps pushing forward, tiny steps, tiny bits, inserting himself carefully into her life. He's been badly treated too, and recognises the need for slow and cautious. He thinks Sky is worth taking time over and he's so full of fabulous ideas. What a great man he is, just perfect for her.
Its a wonderful read, with a mix of humour and pathos, love and loving, and Hayden's special brand of romance, amid a chaotic but close family group. I felt I was part of the mix with them. It was lovely too to see Lucky and James in this book, I do like it when we see how characters from past books are doing.


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These elegant Lace and Jewel Wedding Invitations are perfect for a vintage themed wedding and can also be customised to your colour scheme. Please contact us using our Bespoke Wedding Stationery Enquiry Form and we will get back to you.


text:
APC magazine
TechLife magazine
Back to school deals
PS5 stock
Apple's iPhone sales shortfall could cost it $9 billion in revenue
By Matt Swider published 2 January 19
China, battery replacement and fewer carrier subsidies blamed
Apple isn't selling as many iPhones as it had previously forecast, according to revised first quarter projections, and it's a big enough shortfall to lower its revenue estimates by billions of dollars.
The company lowered its guidance by $9 billion (about £$7.14b, AU$12.9b), going from a projected high of $93 billion (about £74b, AU$133b) to a new estimate of $84 billion (about £67b, AU$120b).
The blame? "Lower than anticipated iPhone revenue, primarily in Greater China," according to a letter by Apple CEO Tim Cook. He cited the slowing of the Chinese economy and US-China trade tensions.
Apple stock was briefly halted and the share price fell over 10 points in after hours trading. If there's any good news, it's that Cook noted that categories outside of the iPhone, including Mac, iPad wearables and services, actually grew by 19%.
Other reason for iPhone weakness
It's not all China. Other factors for weak iPhone performance include the strength of the US dollar and Apple's temporarily price cut for its replacement battery service in older iPhones. That deal ended on December 31.
Cook also cited the fact that there are fewer carrier subsidies in a CNBC interview. In the US, for example, a new iPhone isn't $199 on contract anymore. That changed a while ago, but it's just hitting some new upgraders for the first time.
"There aren't as many subsidies as their used to be from a carrier point of view," said Cook. "That didn't all happen yesterday, but if you were out of the market for two to three years and you come back, it looks like that to you."
The next official Apple earning conference call is expected to happen toward the end of this month, on Tuesday, January 29. We'll have the final numbers and more words from Cook and company then.
Without iOS 12.1.2 on iPad, some users can't import iCloud backups
Matt Swider
Matt Swider is Founder and Editor-in-Chief of The Shortcut.com. Formerly TechRadar's US Editor-in-Chief, he began his tech journalism career all the way back in 1999 at the age of 14. He's tested over 1,000 phones, tablets and wearables and commands a Twitter account of 1m+ followers. Matt received his journalism degree from Penn State University.
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Samsung Galaxy Z Fold 4 needs to up its game to compete with other foldable phones
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Part 5 of 5 in the Teen Beat 26 Anniversary Showcase: The dramatic conclusion to Teen Beat Week here at TSOI!!
The flagship band of Teen Beat records, Unrest was the headliner to each of the six 26th Anniversary shows. Unrest's first cassette came out in 1985 (!?!) but hit their stride in the early '90s and whose sound can be heard in the indie-verse everywhere. I read somewhere that Unrest's Imperial f.f.r.r (1992) provided a nice guitar/drum razor precision counter-point to the ramshackle guitar/musicianship of early '90s slop guitar ala Pavement.
These days you can hear little aspiring Mark Robinson's wrecking their pick-hands trying to out jangle the pro. Good luck with that.
Well I hope you liked Teen Beat Week as much as we did.


text:
Q: Is the difference between "confidante" vs. "confident" the gender of the subject? I observe that the word confidante has circa 3,2001 hits on The New York Times. And I also observe that on the same newspaper the word confident has ca. 148,000 hits.
3,200 seems to me a remarkable number. As is well known, confidante is used only for women, while confident is used both for men and women.
So the question is: Does this represent a persistent tendency to preserve sexism in language? Or are there cases in which confidante is preferable, for whatever reason?
1Searching from other countries could have different results.

A: These are different words.
Confidante is a noun meaning 'a person in whom one confides'. Both this, the feminine form, and the masculine form confidant without the feminine -e ending have been borrowed from French, in which language it derives from a participle. English preserves the gender distinction, which is not, I think, particularly sexist: the words are generally used of particular relationships in which it may be significant to distinguish male and female friends. It is, at any rate, no more sexist than the distinction between boyfriend and girlfriend.
Confident is an adjective, meaning 'sure of oneself' when used without modification, or sure, trusting of [another]. It is derived directly from Latin. Like other English adjectives, it is not inflected


text:
Q: Converting MySQL function to Redshift I'm just starting to learn Redshift and I'm trying to convert a function used on a MySQL server to work in Redshift. I haven't had any luck so far, hoping the stackoverflow wizards can provide direction.
CREATE FUNCTION SS_Ans_to_Num(answer VARCHAR(1000))
 RETURNS INT
 BEGIN
declare answer_num int(11);

if answer = 'No' then SET answer_num = 0;
elseif answer IN ('N/A – I did not receive this training', 'N/C - No recibí esta capacitación') then SET answer_num = 0;
elseif answer IN ('Strongly Disagree', 'Totalmente en desacuerdo') then SET answer_num = 1;
elseif answer IN ('Poor','Mala') then SET answer_num = 1;
elseif answer IN ('Disagree','En desacuerdo') then SET answer_num = 2;
elseif answer IN ('Fair','Regular') then SET answer_num = 2;
elseif answer IN ('Neutral','Neutro') then SET answer_num = 3;
elseif answer IN ('Agree','De acuerdo') then SET answer_num = 4;
elseif answer IN ('Very Good','Muy buena') then SET answer_num = 4;
elseif answer IN ('Yes','Sí') then SET answer_num = 5;
elseif answer IN ('Strongly Agree','Totalmente de acuerdo') then SET answer_num = 5;
elseif answer IN ('Excellent','Excelente') then SET answer_num = 5;
else SET answer_num = 0;
end if;

return (answer_num);

END;

I thought something along these lines would work, but it doesn't:
CREATE FUNCTION SS_Ans_to_Num(answer VARCHAR(1000))
 RETURNS INTEGER
 STABLE
 AS
  $$
   CASE
    WHEN answer = 'No' THEN SET answer_num = 0;
    WHEN answer IN ('N/A – I did not receive this training', 'N/C - No recibí esta capacitación') then SET answer_num = 0;
    WHEN answer IN ('Strongly Disagree', 'Totalmente en desacuerdo') then SET answer_num = 1;
    WHEN answer IN ('Poor','Mala') then SET answer_num = 1;
    WHEN answer IN ('Disagree','En desacuerdo') then SET answer_num = 2;
    WHEN answer IN ('Fair','Regular') then SET answer_num = 2;
    WHEN answer IN ('Neutral','Neutro') then SET answer_num = 3;
    WHEN answer IN ('Agree','De acuerdo') then SET answer_num = 4;
    WHEN answer IN ('Very Good','Muy buena') then SET answer_num = 4;
    WHEN answer IN ('Yes','Sí') then SET answer_num = 5;
    WHEN answer IN ('Strongly Agree','Totalmente de acuerdo') then SET answer_num = 5;
    WHEN answer IN ('Excellent','Excelente') then SET answer_num = 5;
    ELSE SET answer_num = 0;
  END CASE;

RETURN (answer_num);

$$ LANGUAGE SQL

A: The correct syntax would be:
CREATE OR REPLACE FUNCTION SS_Ans_to_Num(TEXT)
 RETURNS INTEGER
 STABLE
 AS
  $$
   SELECT CASE
    WHEN $1 = 'No' THEN 0
    WHEN $1 IN ('N/A – I did not receive this training', 'N/C - No recibí esta capacitación') THEN 0
    WHEN $1 IN ('Strongly Disagree', 'Totalmente en desacuerdo') THEN 1
    WHEN $1 IN ('Poor','Mala') THEN 1
    WHEN $1 IN ('Disagree','En desacuerdo') THEN 2
    WHEN $1 IN ('Fair','Regular') THEN 2
    WHEN $1 IN ('Neutral','Neutro') THEN 3
    WHEN $1 IN ('Agree','De acuerdo') THEN 4
    WHEN $1 IN ('Very Good','Muy buena') THEN 4
    WHEN $1 IN ('Yes','Sí') THEN 5
    WHEN $1 IN ('Strongly Agree','Totalmente de acuerdo') THEN 5
    WHEN $1 IN ('Excellent','Excelente') THEN 5
    ELSE 0
   END

  $$ LANGUAGE SQL

The syntax is similar to an SQL statement rather than a procedural language. If you wish to use a more programmatic language, use language plpythonu instead of language SQL.
See: CREATE FUNCTION documentation


For more than 20 years the friendly staff at Goldfields Nissan have been selling and servicing new and used vehicles in the Kalgoorlie, Goldfields and the surrounding region.
Our new state of the art modern showroom and large display areas allow customers to casually browse the great range of Nissan Vehicles.
Goldfields Nissan actively supports our local community, charities and sporting clubs.
Our staff are dedicated to providing you with the best of care at all times, so whether you are buying, selling or leasing come and experience the difference at Goldfields Nissan.


text:
Planting For the Future
August 30, 2016–Camille Tremblay Beaulieu
Habitat•Nature Reserves
Fritillary butterfly, Willoughby Nature Reserve
On August 19th, I joined a group that was setting off on a tree-planting mission at Ontario Nature's Willoughby Nature Reserve, near Caledon. Armed with shovels and enthusiasm, we were determined to plant 220 trees in just five hours.
The forest was beautiful, cool and quiet as we got to work under the guidance of Ontario Nature's Smera Sukumar and Stephanie Muckle. Leaders from Credit Valley Conservation's Conservation Youth Corps also lent a hand. Our goals for the day were to reduce erosion near Silver Creek, which flows through the reserve, and to improve the surrounding forest's overall health. We also wanted to connect our 25 youth participants with the Willoughby Nature Reserve to deepen their love of the natural world.
Planting native trees at Willoughby Nature Reserve © Camille Tremblay-Beaulieu
Prior to becoming a part of Ontario Nature's system of nature reserves, much of the Willoughby property was cleared and actively farmed. In subsequent years, it was replanted with uniform rows of white, red and Scot's pine trees. When Ontario Nature acquired the property, its forest had few native trees, and was vulnerable to the harmful effects of invading species. Planting a variety of native trees will help improve the reserve's overall ecological health and long-term resilience.
Thanks to the enthusiastic hard work of our youth volunteers, we were able to plant all 220 trees in randomized patterns in just three hours. Muddy, sweaty and happy we had accomplished the seemingly impossible.
We spent the rest of the afternoon hiking and clearing the trail, with 25 pairs of feet patting down a path for others to follow in their explorations of this special nature reserve.
Tree planting at Willoughby Nature Reserve © Camille Tremblay
We'd like to thank the Great Lakes Guardians Community Fund for their support in making this event possible. Visit Ontario Nature's website to learn more about Willoughby Nature Reserve.
Camille Tremblay Beaulieu was an intern at Ontario Nature, a past graduate in Zoology at Laurentian University and a former student of Environmental Visual Communication at the Royal Ontario Museum through Fleming College. She is passionate about photography, conservation and travelling.


Sen. Marco Rubio, R-Fla., talks during a news conference at the Capitol in Washington on Dec. 17, 2014.
Florida Republican Sen. Marco Rubio said Wednesday that his family is on board if he decides to run for president, but he has yet to make up his mind about whether to seek the White House.
Speaking at a breakfast for reporters organized by the Christian Science Monitor, Rubio said he has spoken and emailed with Florida Gov. Jeb Bush since his onetime political mentor announced he would "actively explore" a presidential bid last month. Rubio declined to describe those conversations, but said Bush's decision would not impact his.
"I think Jeb Bush is going to be a very credible candidate," Rubio said, adding that he could "easily" raise more than $100 million to win the Republican nomination.
Rubio's comments come as he is set to meet with donors in Florida this weekend before flying out to California to attend a gathering organized by the billionaire industrialist Koch brothers.
Rubio declined to directly criticize Iowa Republican Rep. Steve King's comments Tuesday evening, in which King called a guest of First Lady Michelle Obama to the State of the Union a "deportable" because she was brought to the U.S. illegally as a child.
Responding to arguments from many Republicans that the next president should be a governor to draw a contrast with President Barack Obama, Rubio quipped: "If I was a governor I'd make the same argument too." But he argued that a grasp of national security issues is more important than executive experience.
"I believe that the central obligation of the federal government… is providing for the national security," he said. "The next President of the United States needs to be someone who has a clear vision of American's role in the world.


The idea of leather nails may sound two ways to a typical, relatively in-the-know beauty consumer: gaudiness at its most fashionable, or awesome-ness at its most fashionable. Nails Inc. have had leather nails for some time now as Alexa Chung wore their original Leather Effect kit, complete with skull and bones decals, to the British Fashion Awards in December 2012. The only-polish version that's also now available in the U.S. in five shades (more choices in the U.K.), is meant to mimic the look of leather as opposed to actual leather applied to your nails.
The colors are great in the bottle. Pastel yet potently colored, and the coral is unexpected. Leather in other colors, even in real life, is always exciting and different. The actual delivery though of the effect is lackluster. It's the kind of special effect that you have to see up close than preferably just a few persons away. Hues wise, the effect looks best in, naturally, the black shade, where the matte finishing helps to make the slightly bumpy texture noticeable. In the other colors, you can barely tell.
The attempt at bringing a much desired nail art detail accessibly is cause for kudos, but it does fall flat, especially when Chung's version really did appear as an leather applique. At an affordable $11, this look may be best for longer nails, where on shorter nails, it won't be as eye-catching.


text:
Shahrekord University of Medical Sciences (SKUMS) is a university in Shahrekord, the capital of the Chaharmahal and Bakhtiari Province of Iran.

Educational hospitals

 Ayatollah Kashani Hospital
 Hajar Hospital
 Vali-Asr Hospital

See also
 Higher education in Iran
 Chahar Mahal and Bakhtiari
 Shahrekord County

External links
Official website

Universities in Iran
Medical schools in Iran
Education in Chaharmahal and Bakhtiari Province
Buildings and structures in Chaharmahal and Bakhtiari Province

1986 establishments in Iran
Educational institutions established in 1986

redpajama_set_name: RedPajamaWikipedia


A Burberry rucksack styled in 100% polyamide signature vintage check with calf leather trim and polished metal hardware. This rucksack features a main compartment, 3 exterior zip pockets, 1 interior slip pocket, wall zip pocket, 2 adjustable shoulder straps, Buckle, press-stud and drawcord closures. DiMen'sions: 8.7 x 5.5 x 13".Made in Italy.


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As we all know, every artwork Van Cleef & Arpels is like a poetic picture scroll, which can always bring you beautiful and dreamy reverie.
This pair of Van Cleef & Arpels Perlee earrings is a great choice for a stylish woman. The frame of the ear-studs is surrounded by delicate beads, making them appealing and playful. At the same time, The circular charm is covered by costly rhinestone, showing the nobility temperament and delicate characteristic of the lady perfectly.
Our knock-off jewelry not only has a more attractive price than the real one but also has excellent quality to ensure a pleasant purchase experience for you.


text:
Cleer Audio ENDURO ANC Review
8th January, 2023 Gary Teasdale
Bassy but balanced native audio tuning
Stylish and premium appearance
Ease of use and simple connectivity
Outstanding battery life
Excellent ANC and Ambient modes
Great Bluetooth range
Mic issues on Discord (PC)
Relatively high Bluetooth latency
The Cleer Audio Enduro ANC has outstanding audio and even better battery life
Manufacturer: Cleer Audio
Model: Enduro ANC
Type: Bluetooth 5.2 Wireless Stereo Headphones
Price: £149.99 MSRP ($129.99 US)
Supported Platforms – Bluetooth: PC, Mac, Switch (Docked), Mobile
Supported Platforms – Wired (3.5mm): PS4/5. Xbox One/Series X|S, Mobile, Switch
Reviewed on: Windows PC, Samsung mobile/tablet
Supplied by: Cleer Audio
The Cleer Audio Enduro ANC surpassed all of my expectations. The performance itself wasn't that surprising; Cleer Audio are renowned for exceptional sound quality, and the Enduro ANC delivers on that front. What was surprising was the way the Enduro ANC brings together so many desirable features at such an affordable price point. When you can make favourable comparisons against competitors which are often in the £300+ price range, you know you've got something special.
I'll admit to being spoilt when it comes to headphones, as I have the Audeze Penrose X for my gaming needs, and when I want the ultimate in fidelity, I switch to the Audeze LCD-GX. That performance comes at a substantial price, though (£300 and £850, respectively). Although £149 isn't exactly cheap, the Cleer Enduro ANC represents amazing value considering the features on offer; tight and clean audio, effective active noise cancellation, an eye-catching aesthetic, and outstanding battery life.
I've been using these headphones for several weeks now, and they have swiftly become my go-to headphones for gaming, movie-watching and commuting. My favourite thing is the simplicity of use. You can pair two Bluetooth devices, and they reliably reconnect without having to dive into any settings. I also never have to worry about battery life. In three weeks of use, I've only needed to charge them twice. That includes when I accidentally left them powered on in my bag all day, which only dropped the battery by 12% over ten hours (they will auto power off, but I'd turned this off in the app – d'oh).
Enduro ANC Headphones
USB-C to 3.5mm audio cable
Twin TRS airplane adapter
Quick-start guide/user manual
Soft-lined canvas carrying pouch
With a navy blue finish accentuated by copper-coloured trim and earth-toned accents, the Enduro ANC looks stunning. Although the majority of the build is plastic, the matte finish looks almost metallic, and it gives the Enduro ANC a premium vibe that I adore. There's a little bit of creaking along the headband when twisted, but this isn't a natural pressure that would be exerted on them, and during regular use, there's no unwanted noise whilst wearing them.
The Enduro ANC is notably more compact than many headphones and headsets I've tested, which makes them ideal for taking on the go. They can also be folded via a neatly concealed hinge, making it easy to pop them into your bag when not in use.
Despite the compact size, these are extremely comfortable headphones. Supremely well-judged clamping force holds the earcups close without unwanted pressure. The Enduro ANC held up well with everyday use, though they may be ever so slightly loose if you plan to go jogging. They also have a discreet pivot mechanism and can be swivelled fully flat if you want to dangle them around your neck. The swivel function is great for storage, but they turn inwards slightly, which was a little uncomfortable around my throat.
The supple leatherette-coated earpads feel great, creating a solid seal around your ears, and they are filled with quietly squishy foam (there's nothing worse than crinkly earcups). The leatherette isn't very breathable, which can cause your ears to get warm with prolonged use, but it provides excellent passive noise cancellation which in turn improves active noise cancellation. I found the spacing inside the cups to be sufficient to cover my ears, but it's a close fit; If you have larger ears it's possible they could be a little snug.
Features and controls
From left to right in the image below; Volume up, power/multi-function, volume down, ANC button
Cleer Audio has done a fantastic job of simplifying the controls on the Enduro ANC. Located on the left ear cup is a simple USB-C port for charging or connecting the USB-C to 3.5mm TRRS audio cable. Slightly higher on the earcup, and exactly where your thumb lands if you reach up to them, are the control buttons.
The larger single button cycles between ANC, ambient mode and ANC off. Pressing and holding this button switches on ambient mode and mutes the sound for as long as you hold it. It's a useful idea in theory, but in the situations where I'd need it, I found it easier to either pause my music or just move an earcup aside momentarily. Additionally, a double-press of this button will summon your device's voice assistant.
The bank of three buttons comprises a central power/multi-function button flanked by volume up and down buttons. The top of the power button is significantly pronounced, which I found immensely handy – with my hood up in the pouring rain, I was easily able to locate and operate the Enduro ANC through a thick layer of fabric, allowing me full control without getting my phone and/or headphones wet.
It may seem limited with so few buttons, but the Enduro ANC is fully featured and a breeze to operate. Holding the power button turns the headphones on and off, a single press will play/pause audio (and video), double pressing skips to the next track and a triple press goes back a track. A double press will also answer/hang up calls (and pause your music), and if you aren't feeling sociable, a long press will reject any incoming call. It's simple but very effective.
In the Enduro ANC's out-of-the-box state, you get the flat (but still excellent) audio EQ. That's not the end of the story, though – if you have an Android or iOS mobile or tablet, you can download the Cleer+ app, which allows you to choose from a range of preset audio profiles or configure custom profiles, and there's an ambient/ANC slider that lets you fine-tune the level of ambient noise the Enduro ANC lets through.
You can also adjust the automatic power-off timing via the app, and any changes will be stored in an onboard profile and applied to whichever device you're using. There's no way to store multiple profiles and switch between them without the app, but at least your current profile and setup remain persistent, so it's not likely to be an issue for many.
The app unfortunately isn't available for PC, but if you're buying £149 Bluetooth headphones I'm going to go out on a limb and say the overwhelming majority at least have access to a mobile device to configure the Enduro on. You can also connect to your PC and phone simultaneously, and as long as your mobile isn't playing audio, you can hear and customise the PC audio in real time, which I found handy when making a universal audio profile for my needs.
Connecting your devices is super easy. When you power on the Enduro ANC, it enables Bluetooth pairing mode and will appear as Cleer ENDURO ANC. If you have Bluetooth 5.2, it's even easier, as it will send a notification to your phone/PC that a new device is available for connection. The Enduro ANC can store two devices in its auto-pair memory, but should you add one by mistake or need to change them, you can (very) long-press the power button and wipe the onboard memory.
(Although you can connect to two devices at once, you can't playback audio from both simultaneously. In my testing, I found the Enduro ANC will prioritise a mobile connection over PC.)
Something I didn't expect is the Enduro ANC's party trick. If NFC is enabled on your phone, you can simply hold it up to the left earcup and it will automatically enable Bluetooth and connect to the headphones, and also disconnect with a tap after use.
The Bluetooth range is exceptional, and I had to walk out of the house and to my front gate (about 8 metres through walls) before it started dropping the signal from my PC. I've tested other Bluetooth headsets that struggle to maintain the signal while I walk to the fridge, so this coverage is hugely impressive.
In addition to Bluetooth, the Enduro ANC also supports 3.5mm jack connections, which also enables high-res audio support and reduces latency. The latency reduction is desirable if you're playing competitive games, as the default Bluetooth latency is around 200ms, which is quite high. With a wired connection this is reduced to around 75ms, which is much better.
As for the high-res audio support, I'll admit that I didn't notice a huge difference between my 320 Kbps mp3 and my WAV files. With lesser headphones, there's a big difference at maximum volume, as this is where distortion usually comes into play, but the audio is so "cleer" <ahem> that any difference is negligible.
The native sound quality is outstanding, and even though some people (wrongly) believe you need at least 50mm drivers to get that rumbly and powerful bass response, Cleer's patented 40mm ironless drivers excel, producing powerful bass and sublime audio that extends across the entire range of sound.
Although the natural audio tuning is described as flat, it's anything but: Bass is remarkably well defined, able to produce powerful and deep tones, as well as punchy accents, without muddying the sound higher up the bands; Midrange audio is also very well portrayed. Audio has a full body, with great projection and presence. The separation between bass and mids is excellent, allowing you to clearly pick out individual instruments in songs and audio cues in games alike; Treble is relatively gentle compared to the powerful bass and mids, but it remains bright and sparkly, with ample clarity and a definite and pleasing crispness.
With all of these audio elements combined, the result is symphonic. Even at low volumes, the sound possesses warmth. You also get an unexpectedly wide soundstage, with crystal clear placement and projection. It's difficult to describe if you haven't experienced decent headphones, but cheap headphones can make the audio sound like it's coming from inside your head, or right next to your ears, at best. Great headphones, like the Enduro ANC, move the audio around you, making the audio sound spacious.
Closed-back headphones can suffer from excess reverb, but thanks to the restrained yet punchy 40mm drivers, there is no unwanted reverberation. Bass from explosions in games and movies still rumbles and has an enjoyable persistence, but this doesn't come at the expense of smothering other frequencies.
For just £149, the Enduro ANC produces sound that will please all but the most anal of audio enthusiasts. When I said these have become my go-to headphones, I meant it – PC gaming, movies, TV shows, music – pretty much everything. Yes, I have better headphones in the LCD-GX, but they have a cumbersome cable and require a separate amp. My Penrose X are arguably better all-rounders (that I still use for console gaming), but I don't like wearing them out because they are large and don't look as good as the Enduro, plus the battery life is "meh" in comparison. If you can find better headphones at this price point, buy 'em!
ANC – Active Noise Cancellation
This isn't the most effective noise cancellation I've tried – the Nuraphone virtually silences everything around you – but the Enduro ANC does a great job at neutralizing a lot of annoying background noise like traffic and the white noise when in a city centre. However, if you've got your music cranked up, then the tight seal, ANC and clear audio at high volumes removes almost all background noise and enables you to immerse yourself in your audio, blissfully removed from the bustle around you.
As with many ANC headphones, wind can be problematic. Strong winds blowing into/across the microphones create noise similar to that heard when you blow into a regular microphone, and it's audible above your music. A decent hood over your head will fix this, but if you don't have one, you'll have to switch ANC off if the weather is bad.
Ambient mode, the flip side to ANC, works brilliantly. I have it cranked all the way up and it allows you to hear everything around you. The stereo effect isn't as precise as it is for music, and with some sounds, it's hard to pinpoint whether they are in front or behind you, but I could clearly hear traffic approaching, conversations of passers-by, my footsteps, the click of my mechanical keyboard, my fingers gently drumming on my desk – essentially everything. The pitch is a little higher than the real thing and there's a bit of background static hiss, but overall it's one of the best ambient audio implementations I've heard, and best of all, it has little impact on the sound quality.
Microphone quality
I'm not going to spend long on the mic quality, as the dual mics are perfectly serviceable for basic voice calls. For gaming, I use the 512 Audio Limelight microphone (fantastic mic, check out our review), and it pairs very well with these headphones, but I'd have trouble recommending the Enduro ANC in place of a gaming headset. This isn't necessarily because of the quality, as the audio is clear and a reasonable representation of my voice, though it is lacking a little bass. The biggest issue is that the Enduro ANC picks up quite a bit of ambient noise and there is no physical mute button. I also had issues with Discord (PC) not picking up any sound from the microphone. Sound was definitely being picked up, as detected when troubleshooting via Windows settings, but for the life of me I couldn't get Discord to work (although I had no issues with the Discord mobile app – go figure?)
These are headphones, though, and not a gaming headset, so I can't in good faith mark them down for this fact, but like the Bluetooth latency, it's something to be aware of if you're considering buying these headphones.
Last but most definitely not least, we come to the battery life – Battery life is rated at 60 hours with ANC and Bluetooth enabled, which sounds like a bold claim, but I'll be damned if they don't excel in this regard. I've been using these headphones for 3 weeks, and I've charged them three times, and not with less than 30% remaining.
For the first week, I recorded my usage as accurately as possible under natural conditions, and I got to 50 hours (over 7 hours per day) with 33% battery remaining; I didn't always have ANC enabled, and the volume was as low as 35% when watching TV whilst working, but that works out as 75 hours of battery life.
For the second week, I went balls to the wall, and cranked them up with ANC switched on blasting SOAD on repeat (for research, of course), and achieved 50% battery remaining after 25 hours of usage.
This is exceptional, and easily the best battery life of any set of headphones or headset I've used (Cleer also make headphones with 100-hour battery life, too, if this isn't enough for you!) The 800 mAh battery sounds small, but believe me, it's more than enough to keep you powered up for days on end.
Charging is arguably quite slow, taking roughly 3 hours from 0-100%, but when you can easily get a week of use from a single charge, it's no big deal finding a couple of hours to charge your headphones.
Even without any of the extra features, the Enduro ANC is a great set of headphones for the price. The native EQ has pleasing amounts of bass, and the sound separation and clarity are exceptional. Factor in all of the quality-of-life features such as the app and effective audio EQ, the ease of use, incredible battery life, effective noise cancellation and brilliant ambient mode, and the Enduro ANC is almost unbeatable. It also looks pretty damn good, too.
In terms of gaming, the latency is quite high and the microphones aren't quite up to what gamers expect, but if you have a separate microphone I'd still recommend the Enduro ANC, especially considering how well it works as an all-rounder.
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Gary Teasdale
Co-owner and Editor in Chief of TGA, Gary has been playing since the '80s. The only thing he likes more than games is gaming hardware (oh yeah, and his children). His questionable claims to fame are his gamerscore of over 370,000, and being the 1994 Hollywood Bowl Mortal Kombat 2 winner-stays-on champion, racking up an unbeaten 52 consecutive wins, with no losses.
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