text:
\section{Introduction}

In the fifth generation mobile communications, the application of millimeter-wave (mmWave) is vital by the virtue of providing high data rate and large bandwidth \cite{R1,R5,G1}. However, mmWave channel suffers from a severe path loss, and the traditional fully-digital precoding with massive antennas to mitigate this issue is extremely power consuming \cite{ref_plus_5}. Therefore, hybrid precoding has been proposed to achieve the large array gains with the reduced hardware cost and power consumption \cite{liao,ref_plus_10,R4,R6}. By far, existing broadband hybrid precoding schemes usually focus on fully-connected array (FCA), while seldom consider partially-connected subarray (PCS) like fixed subarray (FS) and dynamic subarray (DS). Therefore, hybrid precoding with PCS in broadband channel is an interesting topic to explore.

Most prior work is based on narrowband mmWave channels \cite{OMP,ref_plus_9,mao}. Specifically, a compressive sensing-based
hybrid precoding has been proposed in \cite{OMP}, where the channel sparsity is ingeniously exploited to design hybrid precoding with the aid of orthogonal matching pursuit (OMP) algorithm. Moreover, a constant envelope hybrid precoding scheme is proposed, where two cost-efficient sub-connected hybrid architectures are considered to optimize the hybrid precoding under per-antenna constant envelope constraints \cite{ref_plus_9}. To improve bit-error-rate, an over-sampling codebook-based hybrid minimum sum-mean-square-error precoding is designed \cite{mao}. On the other hand, mmWave channels appear to have the frequency-selective fading, where OFDM is usually adopted to combat the time dispersion channels \cite{lim_feedback,170825,fre_flat}. Specifically, an insightful broadband hybrid precoder based on limited-feedback codebook has been proposed for FCA \cite{lim_feedback}. By exploiting the channel correlation information among different subcarriers, a broadband hybrid precoding is proposed for FS and DS \cite{170825}. Finally, \cite{fre_flat} has theoretically shown the optimality of frequency flat precoding by proving that dominant subspaces of the frequency domain channel matrices of different subcarriers are equivalent. However, this conclusion is based on the purely sparse channels with discrete angles of arrival (AoA) and angles of departure (AoD), while the explicit precoder solution is not provided.

In this paper, we propose a machine learning based hybrid precoding scheme for
mmWave MIMO-OFDM systems with DS, where a shared agglomerative hierarchical clustering (shared-AHC) algorithm is proposed for DS grouping to improve SE performance. First, we propose a PCA-based analog precoder scheme for FS by abstracting the low-dimensional signal space of frequency-flat precoder for given subarray from the high-dimensional signal space of optimal frequency-selective fully-digital precoders using PCA. Besides, the optimality of the proposed PCA-based hybrid precoder design is theoretically proven and verified by simulations. Second, we propose the shared-AHC algorithm inspired by cluster analysis in the field of machine learning for antenna grouping. By implementing shared-AHC algorithm, the SE performance of PCS can be further enhanced for effective antenna grouping adapting to the spatial features of the frequency-selective  channels. Finally, we consider the practical passive/active antennas for EE performance analysis. Simulation results confirm the better spectral efficiency (SE) and energy efficiency (EE) performance achieved by the proposed scheme than existing schemes. Meanwhile, DS has the overwhelming advantage for both active and passive antennas.

\textsl{Notations}: Following notations are used throughout this paper. $\mathbf{A}$ is a matrix, $\mathbf{a}$ is a vector, $a$ is a scalar, and $\mathcal{A}$ is a set. Conjugate transpose and transpose of $\mathbf{A}$ are $\mathbf{A}^H$ and $\mathbf{A}^T$, respectively. The $(i,j)$th entry of $\mathbf{A}$ is $[\mathbf{A}]_{i,j}$, and $[\mathbf{A}]_{i,:}$ ($[\mathbf{A}]_{:,j}$) denotes the $i$th row ($j$th column) of $\mathbf{A}$. Frobenius norm is denoted by $||\cdot||_F$. $|\mathbf{A}|$, $|\mathcal{A}|$, $|\mathbf{a}|$, and $|a|$ are the determinant of a square matrix $\mathbf{A}$, cardinality of a set $\mathcal{A}$, $\ell_2$-norm of a vector $\mathbf{a}$, and modulus of a number $a$, respectively. The $i$th largest singular value of a matrix $\mathbf{A}$ is defined as $\lambda_i(\mathbf{A})$. Additionally,
$\text{blkdiag}(\mathbf{a}_1,\cdots,\mathbf{a}_K)$ is a block diagonal matrix with $\mathbf{a}_i$ ($1\leq i\leq K$) on its diagonal blocks.

\section{System Model}
We consider an mmWave massive 3-dimensional (3D) MIMO system, where both the BS and user employ the uniform planar array (UPA), and OFDM is adopted to combat the frequency-selective fading channels. The BS is equipped with $N_t=N_t^v\times N_t^h$ antennas and $N_t^{\rm RF}\ll N_t$ chains, where $N_t^v$ and $N_t^h$ are the numbers of vertical and horizontal transmit antennas, respectively. The user is equipped with $N_r=N_r^v\times N_r^h$ antennas, where $N_r^v$ and $N_r^h$ are the numbers of vertical and horizontal receive antennas, respectively. We consider the downlink transmission, and the received symbols at the user can be written as \cite{OMP}
\begin{equation}\label{rcv_sig}
\mathbf{r}[k]=\mathbf{W}^H[k](\mathbf{H}[k]\mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}[k]\mathbf{x}[k]+\mathbf{n}[k]),
\end{equation}
where $1\leq k\leq K$ with $K$ being the number of subcarriers, $\mathbf{F}_{\rm BB}[k]\in\mathbb{C}^{N_t^{\rm RF}\times N_s}$, $\mathbf{F}_{\rm RF}\in\mathbb{C}^{N_t\times N_t^{\rm RF}}$, $\mathbf{W}[k]\in\mathbb{C}^{N_r\times N_s}$, $\mathbf{H}[k]\in\mathbb{C}^{N_r\times N_t}$, $\mathbf{x}[k]\in\mathbb{C}^{N_s\times 1}$, and $\mathbf{n}[k]\in\mathbb{C}^{N_r\times 1}$ are the digital precoder, analog precoder, fully-digital combiner, channel matrix, transmitted signal, and noise associated with the $k$th subcarrier, respectively, and $N_s$ is the number of data streams for each subcarrier. Noise $\mathbf{n}[k]$ satisfies $\mathbf{n}[k]\sim \mathcal{CN}(0,\sigma_n^2)$, and transmitted signal $\mathbf{x}[k]$ satisfies $\mathbb{E}[\mathbf{x}[k]\mathbf{x}^H\![k]]\!\!=\!\!\frac{P}{KN_s}$, where $P$ is average total transmit power.

The frequency-domain channel $\mathbf{H}[k]$ can be expressed as $\mathbf{H}[k]=\sum_{d=0}^{D-1}\mathbf{H}_d[d]e^{-j\frac{2\pi k}{K}d}$ \cite{lim_feedback}, where $D$ is the maximum delay spread of the discretized channels, and $\mathbf{H}_d[d]\in\mathbb{C}^{N_r\times N_t}$ is the delay-$d$ channel matrix. We consider the clustered channel model \cite{OMP}, where the channel is composed by $N_{\rm cl}$ clusters of multipaths with $N_{\rm ray}$ rays in each cluster. Thus the delay-$d$ channel matrix can be written as
\begin{equation}\label{H_d}
  \mathbf{H}_d[d]=\sum\nolimits_{i=1}^{N_{\rm cl}}\sum\nolimits_{l=1}^{N_{\rm ray}}p_{i,l}^d[d]\mathbf{a}_r(\phi^r_{i,l},\theta^r_{i,l})\mathbf{a}_t^H(\phi^t_{i,l},\theta^t_{i,l}),
\end{equation}
where $p_{i,l}^d[d]=\sqrt{N_tN_r/(N_{\rm cl}N_{\rm ray})}\alpha_{i,l}p(dT_s-\tau_{i,l})$ is the delay-domain channel coefficient, $\tau _{i,l}$,  $\alpha _{i,l}$, and $p(\tau)$ are the delay, the complex path gain, and the pulse shaping filter for $T_s$-spaced signaling, respectively. Thus the relationship between the frequency-domain channel coefficiency and the delay-domain channel coefficiency is $p_{i,l}[k]=\sum_{d=0}^{D-1}p_{i,l}[d]\exp(-j2\pi kd/K)$. In (\ref{H_d}), $\mathbf{a}_t(\phi^t_{i,l},\theta^t_{i,l})$ and $\mathbf{a}_r(\phi^r_{i,l},\theta^r_{i,l})$ are the steering vectors of the $l$th path in the $i$th cluster at the transmitter and receiver, respectively. In the steering vectors, $\phi^t_{i,l}$ and $\theta^t_{i,l}$ are the azimuth and elevation angles of the $l$th ray in the $i$th cluster for AoDs, and $\phi^r_{i,l}$ and $\theta^r_{i,l}$ are the azimuth and elevation angles of the $l$th ray in the $i$th cluster for AoAs. Therefore, the transmit steering vectors for the UPA at the BS can be expressed as $\mathbf{a}_t(\phi^t_{i,l},\theta^t_{i,l})=
[1\ \ \ \cdots \ \ \ e^{-j2\pi(m\frac{d_h}{\lambda}\sin(\theta ^t_{i,l})\cos(\phi ^t_{i,l})+n\frac{d_v}{\lambda}\sin(\phi ^t_{i,l}))} \ \ \
\cdots \\e^{-j2\pi((N_t^h-1)\frac{d_h}{\lambda}\sin(\theta ^t_{i,l})\cos(\phi ^t_{i,l})+(N_t^v-1)\frac{d_v}{\lambda}\sin(\phi ^t_{i,l}))}]^T/\sqrt{N_t}$ \cite{OMP},
where $\lambda$ is the carrier wavelength, and $d_v$ and $d_h$ are the distances between adjacent antenna elements in vertical and horizontal direction, respectively. Similarly, we can also obtain $\mathbf{a}_r(\phi ^r_{i,l},\theta^r_{i,l})$ with the same form.

The achievable rate of the mmWave MIMO heavily depends on the transmit hybrid precoder, which can be obtained by  solving the following optimization problem \cite{lim_feedback}
\begin{equation} \label{opt}
\begin{aligned}
\max\limits_{\mathbf{F}_{\rm RF},\mathbf{F}_{\rm BB}}&\!\!\sum\nolimits_{k=1}^{K}\!\!\!\!\!\log_2|\mathbf{I}\!+\!\tfrac{1}{\sigma_n^2}\mathbf{H}[k]\mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}[k]
\mathbf{F}_{\rm BB}^H[k]\mathbf{F}_{\rm RF}^H\mathbf{H}^H[k]|
\\\text{s.t. }&\mathbf{F}_{\rm RF}\in\mathcal{F}_{\rm RF},\sum\nolimits_{k=1}^K||\mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}[k]||_F^2=KN_s,
\end{aligned}
\end{equation}
where $\mathcal{F}_{\rm RF}$ is a set of feasible RF precoder satisfying constant-modulus constraint. The coupling between $\mathbf{F}_{\rm RF}$ and $\{\mathbf{F}_{\rm BB}[k]\}_{k=1}^K$ and the constant-modulus constraint of $\mathcal{F}_{\rm RF}$ lead to the challenging hybrid precoder design.

\section{PCA-Based Hybrid Precoder Design for FS}
In this section, we derive hybrid precoders for the PCS, in which only a subset of antennas are connected to each RF chain. Our goal is to design the optimal frequency-flat RF precoder from the fully-digital frequency-selective precoder.
\subsection{Digital Precoder Design}
We first design the digital precoder by fixing the RF precoder. Solving (\ref{opt}) can be difficult due to the coupling of the baseband and RF precoders\cite{lim_feedback}. Therefore, considering $\mathbf{\widetilde{F}}_{\rm BB}[k]=(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{\frac{1}{2}}\mathbf{F}_{\rm BB}[k]$ to be the equivalent baseband precoder,
the equivalent problem of (\ref{opt}) can be expressed as follows
\begin{equation}
\begin{aligned}\label{opt_RF}
\max\limits_{\mathbf{F}_{\rm RF},\mathbf{\widetilde{F}}_{\rm BB}}&\!\!\sum\nolimits_{k=1}^{K}\!\!\!\!\!\log_2|\mathbf{I}\!+\!\tfrac{1}{\sigma_{\rm n}^2}
\mathbf{H}[k]\mathbf{F}_{\rm RF}(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{-\frac{1}{2}}\mathbf{\widetilde{F}}_{\rm BB}[k]
\\&\times\mathbf{\widetilde{F}}_{\rm BB}^H[k](\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{-\frac{1}{2}}\mathbf{F}_{\rm RF}^H\mathbf{H}^H[k]|
\\\text{s.t. }&\mathbf{F}_{\rm RF}\in\mathcal{F}_{\rm RF},\sum\nolimits_{k=1}^K||\mathbf{\widetilde{F}}_{\rm BB}[k]||_F^2=KN_s.
\end{aligned}
\end{equation}

For the optimization problem (\ref{opt_RF}), we first consider the optimal solution of $\{\mathbf{\widetilde{F}}_{\rm BB}[k]\}_{k=1}^K$. Specifically, consider the singular value decomposition (SVD) of $\mathbf{H}[k]$ associated with the $k$th subcarrier as $\mathbf{H}[k]=\mathbf{U}[k]\mathbf{\Sigma}[k]\mathbf{V}^H[k]$, and the SVD of the matrix $\mathbf{\Sigma}[k]\mathbf{V}^H[k]\mathbf{F}_{\rm RF}(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{-1/2}=\mathbf{\widetilde{U}}[k]\mathbf{\widetilde{\Sigma}}[k]
\mathbf{\widetilde{V}}^H[k]$. Therefore, the optimal $\mathbf{\widetilde{F}}_{\rm BB}[k]=[\mathbf{\widetilde{V}}[k]]_{:,1:N_s}\mathbf{\Lambda}[k]$, and thus the optimal baseband precoder $\mathbf{F}_{\rm BB}[k]$ can be expressed as
\begin{equation}\label{F_BB}
\begin{aligned}
\mathbf{F}_{\rm BB}[k]=&(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{-\frac{1}{2}}\mathbf{\widetilde{F}}_{\rm BB}[k]
\\=&(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{-\frac{1}{2}}[\mathbf{\widetilde{V}}[k]]_{:,1:N_s}\mathbf{\Lambda}[k],
\end{aligned}
\end{equation}
where $\mathbf{\Lambda}[k]=(\mu -N_s/[\mathbf{\widetilde{\Sigma}}[k]]_{i,i}^2)^+$ ($1\leq i\leq N_s$, $1\leq k\leq K$) is a water-filling solution matrix, in which $\mu$ satisfies $\sum_{k=1}^K\sum_{i=1}^{N_s}(\mu-N_s/[\mathbf{\widetilde{\Sigma}}[k]]_{i,i}^2)^+=KN_s$.
Then the problem reduces to obtain the optimal solution of $\mathbf{F}_{\rm RF}$ to (\ref{opt_RF}).
\subsection{PCA-Based Precoder Design}
Regarding the transmit hybrid precoder for FS, there are $N_t$ antennas and $N_t^{\rm RF}$ RF chains. For simplicity, we consider the numbers of antennas for different RF chains are identical, and the cardinality of each subset for every antenna group is $N_t^{\rm sub}=N_t/N_t^{\rm RF}$. We define the set of antenna indexes as $\{1,\cdots,N_t\}$, and $\mathcal{S}_r$ as the subset of the antennas associated with the $r$th RF chain, where $\mathcal{S}_r=\{(r-1)N_t^{\rm sub}+1,\cdots,rN_t^{\rm sub}\}$, for $1\leq r\leq N_t^{\rm RF}$. For the FS, the analog precoder $\mathbf{F}_{\rm RF}$ can be written as a block diagonal matrix $\mathbf{F}_{\rm RF}=\text{blkdiag}(\mathbf{f}_{{\rm RF},\mathcal{S}_1},\cdots,\mathbf{f}_{{\rm RF},\mathcal{S}_{N_t^{\rm RF}}})$, where $\mathbf{f}_{{\rm RF},\mathcal{S}_r}\in\mathbb{C}^{N_t^{\rm sub}\times 1}$ is the analog beamforming vector associated with the $r$th subarray for the $r$th RF chain. Defining optimal digital precoder $\mathbf{F}_{\rm opt}[k]=[\mathbf{V}[k]]_{:,1:N_s}$ for $1\leq k\leq K$, the optimal digital precoder can be expressed as
\begin{equation}\label{blk_F_opt}
\mathbf{F}_{\rm opt}^H[k]=\begin{bmatrix}
\mathbf{F}_{{\rm opt},\mathcal{S}_1}^H[k] \
\cdots \
\mathbf{F}_{{\rm opt},\mathcal{S}_{N_t^{\rm RF}}}^H[k]
\end{bmatrix},
\end{equation}
where $\mathbf{F}_{{\rm opt},\mathcal{S}_r}[k]\in\mathbb{C}^{N_t^{\rm sub}\times N_t^{\rm RF}}$. Moreover, we regard the matrix $\mathbf{F}_{\mathcal{S}_r}=\begin{bmatrix}\mathbf{F}_{{\rm opt},\mathcal{S}_r}[1]\ \mathbf{F}_{{\rm opt},\mathcal{S}_r}[2]\ \cdots\ \mathbf{F}_{{\rm opt},\mathcal{S}_r}[K]\end{bmatrix}$
consisting of the optimal precoder of all subcarriers in the $r$th subarray as the data set in the PCA problem \cite{mach_lern}. Additionally, to achieve the stable solution with low complexity for PCA, SVD is applied to the data set matrix $\mathbf{F}_{\mathcal{S}_r}$. This process is detailed in Proposition 1, where its optimality is also verified as follows.
\newtheorem{prop}{\textbf{Proposition}}
\begin{prop}
    For FS, considering $\mathbf{F}_{\mathcal{S}_r}=\begin{bmatrix}
	\mathbf{F}_{{\rm opt},\mathcal{S}_r}[1] &\cdots& \mathbf{F}_{{\rm opt},\mathcal{S}_r}[K]\end{bmatrix}$, the RF precoder $\mathbf{F}_{\rm RF}$ solving problem (\ref{opt_final_eq}) with the subarray analog/digital architecture is given by $\mathbf{F}=\text{\rm blkdiag}(\mathbf{f}_{{\rm RF},\mathcal{S}_1},\cdots,\mathbf{f}_{{\rm RF},\mathcal{S}_{N_t^{\rm RF}}})$, with $\mathbf{f}_{{\rm RF},\mathcal{S}_r}=\alpha_r\mathbf{u}_{\mathcal{S}_r}$, for $r=1,\cdots,N_t^{\rm RF}$, where $\alpha_r\in\mathbb{C}$ and $\mathbf{u}_{\mathcal{S}_r}$ is the right singular vector corresponding with the largest singular value of the matrix $\mathbf{F}_{\mathcal{S}_r}$.
\end{prop}
\begin{proof}
	Following the similar steps of the equations (12)-(14) in \cite{OMP} and defining $[\mathbf{\Sigma}[k]]_{1:N_s,1:N_s}=\mathbf{\Sigma}_1[k]$, the objective function in problem (\ref{opt}) can be approximate as
\begin{equation}
\begin{aligned}
&\sum\nolimits_{k=1}^{K}\log_2|\mathbf{I}+\tfrac{1}{\sigma_n^2}\mathbf{H}[k]\mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}[k]
\mathbf{F}_{\rm BB}^H[k]\mathbf{F}_{\rm RF}^H\mathbf{H}^H[k]|
\\\approx&\!\sum\nolimits_{k=1}^{K}\!(\log_2\!|\mathbf{I}_{N_s}\!\!+\!\!\tfrac{1}{\sigma_n^2}
\!\mathbf{\Sigma}_1^2[k]|
\!-\!(\!N_s\!\!-\!\!||\mathbf{F}_{\rm opt}^H\![k]\mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}[k]||_F^2)\!).
\end{aligned}
\end{equation}
Therefore, the optimization problem (\ref{opt}) is equivalent to the following optimization problem
\begin{equation} \label{opt_final}
\begin{aligned}
\max\limits_{\mathbf{F}_{\rm RF},\mathbf{F}_{\rm BB}}&\sum\nolimits_{k=1}^{K}||\mathbf{F}_{\rm opt}^H[k]\mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}[k]||_F^2
\\\text{s.t. }&\mathbf{F}_{\rm RF}\in\mathcal{F}_{\rm RF},\sum\nolimits_{k=1}^K||\mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}[k]||_F^2=KN_s,
\end{aligned}
\end{equation}
where $\mathcal{F}_{\rm RF}$ is a set of feasible RF precoder satisfying constant-modulus constraint.
The objective function in (\ref{opt_final}) is
\begin{equation
\begin{aligned}
\sum\nolimits_{k=1}^{K}&\!\!\!\!\!\!
||\mathbf{F}_{\rm opt}^H\![k]\mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}[k]||_F^2
\!=\!\!\sum\nolimits_{k=1}^{K}\!\!\!\!\!\!\!\!\text{Tr}(\mathbf{F}_{\rm opt}^H\![k]\mathbf{F}_{\rm RF}\!(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF}\!)\!^{-\!\frac{1}{2}}
\\&\times(\mathbf{\widetilde{F}}_{\rm BB}[k]\mathbf{\widetilde{F}}_{\rm BB}^H[k])
(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{-\frac{1}{2}}\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm opt}[k]).
\end{aligned}
\end{equation}
According to previous work \cite{uni_cons}, unitary constraints offer a close performance to the total power constraint while providing a relatively simple form of solution. To simplify the problem, we consider condition under unitary power constraints instead. Therefore, water-filling power allocation coefficients can be ignored. In detail, the equivalent baseband precoder $\mathbf{\widetilde{F}}_{\rm BB}[k]=[\mathbf{\widetilde{V}}[k]]_{:,1:N_s}$, which means that $\mathbf{\widetilde{F}}_{\rm BB}[k]$ is a unitary or simi-unitary matrix depending on the relationship between $N_s$ and $N_t^{\rm RF}$. When $N_s=N_t^{\rm RF}$, $\mathbf{\widetilde{F}}_{\rm BB}[k]\mathbf{\widetilde{F}}_{\rm BB}^H[k]$ is $\mathbf{I}_{N_s}$. When $N_s<N_t^{\rm RF}$, denoting the SVD of $\mathbf{\widetilde{F}}_{\rm BB}[k]=\mathbf{U}_{\rm BB}[k]\begin{bmatrix}
\mathbf{I_{N_s}} \ \mathbf{0} \end{bmatrix}^T\mathbf{V}_{\rm BB}^H[k]$, thus $\mathbf{\widetilde{F}}_{\rm BB}[k]\mathbf{\widetilde{F}}_{\rm BB}^H[k]=\mathbf{U}_{\rm BB}[k]\text{blkdiag}(\mathbf{I}_{N_s},\mathbf{0}_{N_t^{\rm RF}-N_s})\mathbf{U}_{\rm BB}^H[k]$. Therefore, the solution to the condition when $N_s=N_t^{\rm RF}$ will also suffice the condition when $N_s<N_t^{\rm RF}$. Therefore, the objective function of (\ref{opt_final}) goes down to
\begin{equation}\label{opt_final_eq}
\begin{aligned}
&\text{Tr}(\mathbf{F}_{\rm opt}^H[k]\mathbf{F}_{\rm RF}(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{-\frac{1}{2}}(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{-\frac{1}{2}}\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm opt}[k])
\\=&||\mathbf{F}_{\rm opt}^H[k]\mathbf{F}_{\rm RF}(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})^{-\frac{1}{2}}||_F^2.
\end{aligned}
\end{equation}
For simplicity, we denote $\mathbf{F}_{\rm RF}{(\mathbf{F}_{\rm RF}^H\mathbf{F}_{\rm RF})}^{-1/2}$ as $\mathbf{\bar{F}}_{\rm RF}$. Therefore, $\mathbf{\bar{F}}_{\rm RF}$ can be written into following block diagram matrix
\begin{equation}\label{F_bar}
\mathbf{\bar{F}}_{\rm RF} \!=\!\text{blkdiag}(\mathbf{f}_{{\rm RF},\mathcal{S}_1}|\mathbf{f}_{{\rm RF},\mathcal{S}_1}|^{-1}\!,\cdots,\mathbf{f}_{{\rm RF},\mathcal{S}_{N_t^{\rm RF}}}|\mathbf{f}_{{\rm RF},\mathcal{S}_{N_t^{\rm RF}}}|^{-1}).
\end{equation}
By substituting (\ref{F_bar}) and (\ref{blk_F_opt}) into (\ref{opt_final_eq}), the objective function of the optimization problem can be further expressed as
	\begin{equation}
	\begin{aligned}
	&\sum\nolimits_{k=1}^{K}||\mathbf{F}_{\rm opt}^H[k]\mathbf{\bar{F}}_{\rm RF}||_F^2
	\\=&\sum_{k=1}^{K}||\begin{bmatrix}
	\frac{\mathbf{f}_{{\rm RF},\mathcal{S}_1}\mathbf{F}_{{\rm opt},\mathcal{S}_1}^H[k]}{|\mathbf{f}_{{\rm RF},\mathcal{S}_1}|} &
	\cdots &
	\frac{\mathbf{f}_{{\rm RF},\mathcal{S}_{N_t^{\rm RF}}}\mathbf{F}_{{\rm opt},\mathcal{S}_{N_t^{\rm RF}}}^H[k]}{|\mathbf{f}_{{\rm RF},\mathcal{S}_{N_t^{\rm RF}}}|}
	\end{bmatrix}||_F^2

	\\=&\sum_{r=1}^{N_t^{\rm RF}}\frac{\mathbf{f}_{{\rm RF},\mathcal{S}_r}\mathbf{F}_{\mathcal{S}_r}^H\mathbf{F}_{\mathcal{S}_r}\mathbf{f}_{{\rm RF},\mathcal{S}_r}^H}{|\mathbf{f}_{{\rm RF},\mathcal{S}_r}|^2}.
	\end{aligned}
	\end{equation}
	Therefore, the solution to the optimization problem (\ref{opt_final_eq}) is $\max_{\mathbf{F}_{\rm RF}}\sum_{k=1}^{K}||\mathbf{\bar{F}}_{\rm RF}\mathbf{F}_{\rm opt}^H[k]||_F^2=\sum_{r=1}^{N_t^{\rm RF}}\lambda_1^2(\mathbf{F}_{\mathcal{S}_r})$. The maximum value can only be obtained when $\mathbf{f}_{{\rm RF},\mathcal{S}_r}=\alpha_r\mathbf{u}_{\mathcal{S}_r,1}$, where $\alpha_r$ is an arbitrary complex value, and $\mathbf{u}_{\mathcal{S}_r}$ is the largest singular value of the matrix $\mathbf{F}_{\mathcal{S}_r}$.
\end{proof}

Taking the constraint of RF precoder into account, we can design the RF precoder by solving
\begin{equation}\label{angle_sel}
\begin{aligned}
\mathbf{F}_{\rm RF}&=\text{blkdiag}(\mathbf{f}_{{\rm RF},\mathcal{S}_1},\cdots,\mathbf{f}_{{\rm RF},\mathcal{S}_{N_t^{\rm RF}}})
\\\text{where, }&\mathbf{f}_{{\rm RF},\mathcal{S}_r}=\arg\min_{\mathbf{x},|[\mathbf{x}]_{i,j}|=1/\sqrt{N_{\rm sub}}}||\mathbf{x}-
\mathbf{u}_{\mathcal{S}_r}||^2_F,
\\&\text{for }r=1,\cdots,N_t^{\rm RF}.
\end{aligned}
\end{equation}
With the constant-modulus constraint, the set of possible $\mathbf{f}_{{\rm RF},\mathcal{S}_r}$ is actually a hypersphere in the space of $\mathbb{C}^{N_t\times 1}$, and $\mathbf{u}_{\mathcal{S}_r}$ is a known point in the space of $\mathbb{C}^{N_t\times 1}$. Therefore, the optimization problem in (\ref{angle_sel}) is actually a distance minimization problem. Therefore, the solution is the point on this hypersphere sharing same direction of the know point $[\mathbf{f}_{{\rm RF},\mathcal{S}_r}]_{i}=\sqrt{N_{\rm sub}}e^{j\angle([\mathbf{u}_{\mathcal{S}_r}]_{i})}$.

When the quantization of phase shifters is considered, we assume the quantization bits are $Q$. Therefore, the phase shifters can only be chosen from the following quantized phase set $\mathcal{Q}=\{0,\frac{2\pi}{2^Q},\cdots,\frac{2\pi(2^Q-1)}{2^Q}\}$. Specifically, after obtaining the RF precoder $\mathbf{F}_{\rm RF}$, the quantization process can be realized by searching for the minimum Euclidean distance between $\angle([\mathbf{F}_{\rm RF}]_{i,j})$ and quantized phase from $\mathcal{Q}$.

\section{Shared-AHC Algorithm for DS Grouping}
In Section III, we have found the SE performance of FS heavily depends on $\{\mathbf{F}_{\mathcal{S}_r}\}_{r=1}^{N_t^{\rm RF}}$. This observation motivates us to optimize the antenna grouping $\{\mathcal{S}_r\}_{r=1}^{N_t^{\rm RF}}$ to further improve the SE performance when DS is considered.

The DS problem can be formulated as follows
\begin{equation}\label{pro_dy}
\begin{aligned}
&\max\limits_{\mathcal{S}_1,\cdots,\mathcal{S}_{N_t^{\rm RF}}}\sum\nolimits_{r=1}^{N_t^{\rm RF}}\lambda_1^2(\mathbf{F}_{\mathcal{S}_r})
\\&\text{s.t. }\cup_{r=1}^{N_t^{\rm RF}}\!\mathcal{S}_r\!=\!\{1,\!\cdots\!,\!N_t\},
\ \mathcal{S}_i\!\cap\!\mathcal{S}_j\!=\!\emptyset\text{ for }i\!\not=\!j,\ \mathcal{S}_r\!\not=\!\emptyset\ \forall r.
\end{aligned}
\end{equation}
This optimization problem is a combinational optimization problem, which requires an exhaustive search to reach the optimal solution. To obtain the optimal solution, the number of all possible combinations for exhaustive search can be $\frac{1}{(N_t^{\rm RF})!}\sum_{n=0}^{N_r^{\rm RF}}(-1)^{N_t^{\rm RF}-n}\binom{N_t^{\rm RF}}{n}n^{N_t}$ according to \cite{S_num}, which is a very large number. To illustrate, when $N_t=64$ and $N_t^{\rm RF}=4$, the number of all possible combinations can be up to $1.4178\times 10^{37}$.

Therefore, a low-complexity algorithm need to be develop to solve problem (\ref{pro_dy}). Specifically, we use the Minkowski $\ell_2$-norm \cite{norm} to estimate the square of the singular value of the matrix $\mathbf{F}_{\mathcal{S}_r}$ by
$\lambda_1^2(\mathbf{F}_{\mathcal{S}_r})=\lambda_1(\mathbf{R}_{\mathcal{S}_r})\approx \frac{1}{|\mathcal{S}_r|}\sum_{i=1}^{|\mathcal{S}_r|}\sum_{j=1}^{|\mathcal{S}_r|}|[\mathbf{R}_{\mathcal{S}_r}]_{i,j}|
=\frac{1}{|\mathcal{S}_r|}\sum_{i\in\mathcal{S}_r }\sum_{j\in\mathcal{S}_r}|[\mathbf{R}_F]_{i,j}|$, where $\mathbf{R}_{\mathcal{S}_r}=\mathbf{F}_{\mathcal{S}_r}\mathbf{F}_{\mathcal{S}_r}^H$ and $\mathbf{R}_F=\mathbf{F}\mathbf{F}^H$.

To reduce the complexity while achieve the good SE performance, we consider the antenna grouping from the viewpoint of clustering analysis in machine learning. To be specific, we propose a shared-AHC algorithm as listed in Algorithm \ref{alg:AHC}, which is developed from the AHC algorithm in machine
learning to group the antennas into different subarrays associated with different RF chains. Traditional AHC algorithm is a clustering algorithm that builds a cluster hierarchy from the bottom up. It starts by adding all data to multiple clusters, followed by iteratively pair-wise merging these clusters until only one cluster is left at the top of the hierarchy. The shared-AHC algorithm is different from the traditional AHC algorithm \cite{AHC} in two distinguished aspects. First, the aim of clustering in our antenna grouping problem is to build $N_t^{\rm RF}$ clusters instead of only one cluster in conventional AHC algorithm. Second, the pair-wise merging criterion in the proposed algorithm is ``shared", while the conventional AHC algorithm only considers the target cluster. To further illustrate the ``shared" mechanism, we  introduce the metric of mutual correlation $g(\mathcal{S}_n,\mathcal{S}_m)$ between the cluster $\mathcal{S}_n$ and $\mathcal{S}_m$
\begin{equation}\label{mul_cor}
g(\mathcal{S}_n,\mathcal{S}_m)=\frac{1}{|\mathcal{S}_n||\mathcal{S}_m|}
\sum_{i\in\mathcal{S}_n}\sum_{j\in\mathcal{S}_m}|[\mathbf{R}_F]_{i,j}|.
\end{equation}
In each clustering iteration, we first focus on a cluster $\mathcal{S}_n$, and find a cluster $\mathcal{S}_m$ maximizes $g(\mathcal{S}_n,\mathcal{S}_l)$ among all possible $\mathcal{S}_l$. If the cluster $\mathcal{S}_n$ also maximizes $g(\mathcal{S}_m,\mathcal{S}_l)$ among all possible $\mathcal{S}_l$, we merge $\mathcal{S}_n$ and $\mathcal{S}_m$. Otherwise, the cluster $\mathcal{S}_n$ and cluster $\mathcal{S}_m$ are not merged and algorithm goes into the next iteration. Therefore, our proposed algorithm is featured as ``shared", since two clusters mutually share the maximum correlation in the sense of (\ref{mul_cor}). This process is realized in Algorithm 1.
\begin{algorithm}[htp]
	\caption{Shared Agglomerative Hierarchical Clustering (Shared-AHC) Algorithm for DS Grouping.}
	\label{alg:AHC}
	\begin{algorithmic}[1]
		\renewcommand{\algorithmicrequire}{\textbf{Input:}}
		\renewcommand\algorithmicensure {\textbf{Output:}}
		\Require
		$\mathbf{R}_F$, number of antennas and RF chains $N_t$, $N_t^{\rm RF}$.
		\Ensure
		Grouping result $\mathcal{S}_1,\cdots,\mathcal{S}_{N_t^{\rm RF}}$.
		\State $N_{\rm sub}=N_t$, $\mathcal{S}_i=\{i\}$ for $i=1,\cdots,N_t$
		\While{$N_{\rm sub}>N_t^{\rm RF}$}
		\State $\mathcal{S}_i^0=\mathcal{S}_i$ for $i=1,\cdots,N_{\rm sub}$, $n_{\rm sub}=1$
		\For{$i=1:N_{\rm sub}$}
		\State \textbf{if} $\exists r_0\text{ s.t. }\mathcal{S}_i^0\in\mathcal{S}_{r_0}$ \textbf{then continue}
		\State \textbf{else if} $i=N_{\rm sub}$ \textbf{then} $\mathcal{S}_{n_{\rm sub}}=\mathcal{S}_i^0$
		\State \textbf{else}
		\State \qquad $j=\arg\max\limits_{l\in\{i+1,\cdots,N_{\rm sub}\}}g(\mathcal{S}_i,\mathcal{S}_l)$
        \State \qquad $i^0=\arg\max\limits_{l\in\{1,\cdots,N_{\rm sub}\}\setminus\{j\}}g(\mathcal{S}_j,\mathcal{S}_l)$
		\State \qquad \textbf{if} $i=i^0$ \textbf{then} $\mathcal{S}_{n_{\rm sub}}=\mathcal{S}_i\cup\mathcal{S}_j$
		\State \qquad \textbf{else} $\mathcal{S}_{n_{\rm sub}}=\mathcal{S}_i$
		\State \qquad\textbf{end if}
		\State \textbf{end if}
		\State $n_{\rm sub}=n_{\rm sub}+1$
		\EndFor
		\State $N_{\rm sub}^0=n_{\rm sub}-1$
		\State \textbf{if} {$N_{\rm sub}^0<N_r^{\rm RF}$} \textbf{then} $\mathcal{S}_i=\mathcal{S}_i^0$ for $i=1,\cdots,N_{\rm sub}$
        \State \qquad\textbf{break}
		\State \textbf{else} $N_{\rm sub}=N_{\rm sub}^0$
		\State \textbf{end if}
		\EndWhile
		\If{$N_{\rm sub}>N_t^{\rm RF}$}
		\State Sort $\mathcal{S}_i$ according to the ascending order of cardinality
		\For{$i=1:N_t^{\rm RF}-N_{\rm sub}$}
		\State $j=\arg\max\limits_{l=\{N_t^{\rm RF}-N_{\rm sub}+1,\cdots,N_{\rm sub}\}}g(\mathcal{S}_i,\mathcal{S}_l)$
		\State $\mathcal{S}_i=\mathcal{S}_i\cup\mathcal{S}_j$
		\EndFor
		\State Rearrange the subscript to guarantee that the order of subscripts are from 1 to $N_t^{\rm RF}$
		\EndIf
	\end{algorithmic}
\end{algorithm}
\section{Energy Efficiency Analysis}
The implementation of PCS not only reduces
the hardware complexity, but also improves the EE.
In this section, we analyze the EE of the
designs. Define the EE as $\eta =RB/P$,
where $B$ is the bandwidth of the channel, and $P$ is the total power consumption of the system.

Different connection patterns between
the phase shifters and antennas can influence
the power consumption. Because the number of phase shifters is different in different connection patterns. In this system, FCA use up to $N_tN_t^{\rm RF}$ phase shifter for each RF chain
connecting to every antennas. While the PCS use $N_t$ phase shifters.

Furthermore, the different antenna architectures should also be taken into account regarding the  total power. Specifically, we consider the hybrid MIMO system using passive antennas and active antennas as shown in Fig. 10 of \cite{ant_str}. Both of them consist electronic components such as digital-analog convertors (DAC), power amplifiers (PA), local oscillators (LO), and mixers etc. The main difference between active and passive antenna architecture lies in the number of the PAs. In passive antennas, the number of PAs is the same as that of the RF chains. While for active antennas, the number of PAs is the same as that of antennas. This difference can lead to different power consumption because the PAs are heavily power-consuming. Thus we will analyze the power consumption of the two different antenna architectures, respectively.

Given the above antenna architecture, the power consumption for FCA and PCS are respectively
$P_{\rm FCA}^p=N_tN_t^{\rm RF}P_{\rm PS}\!+\!N_t^{\rm RF}(P_{\rm DAC}\!+\!P_{\rm mix}\!+\!P_{\rm PA}\!+\!P_{\rm LO})$ and $P_{\rm PCS}^p=N_tP_{\rm PS}\!+\!N_t^{\rm RF}(P_{\rm DAC}+P_{\rm mix}\!+\!P_{\rm PA}\!+\!P_{\rm LO})$. By contrast, the power consumption for FCA and PCS with active antenna architecture are $P_{\rm FCA}^a=N_tN_t^{\rm RF}P_{\rm PS}\!+\!N_tP_{\rm PA}\!+\!N_t^{\rm RF}(P_{\rm DAC}\!+\!P_{\rm mix}\!+\!P_{\rm LO}\!)$ and
$P_{\rm PCS}^a=N_tP_{\rm PS}\!+\!N_tP_{\rm PA}\!+\!N_t^{\rm RF}(P_{\rm DAC}\!+\!P_{\rm mix}\!+\!P_{\rm LO})$.
Moreover, according to the antenna architecture for fully-digital (FD), the power consumption is
$P_{\rm FD}=N_t(P_{\rm PA}+P_{\rm DAC}+P_{\rm mix}+P_{\rm LO})$.
Additionally, the power consumption of electronic components in the three architectures are phase shifter $P_{\rm PS}=30$ mW \cite{27}, DAC $P_{\rm DAC}=200$ mW \cite{27}, mixer $P_{\rm mix}=39$ mW \cite{39}, LO $P_{\rm LO}=5$ mW \cite{27}, and PA $P_{\rm PA}=138$ mW \cite{36}.
\section{Simulations}
In this section, we investigate the SE and EE performance for the hybrid precoder design. For the channel model, we adopt Dirac delta function as the pulse shaping filter and a cyclic prefix with the length of $D=64$. The number of subcarriers is $K=512$. The transmission bandwidth is $B=500$ MHz. We consider that the path delay is uniformly distributed in $[0,DT_s]$ ($T_s=1/B$ is the symbol period). The number of the clusters is $N_{\rm cl}=8$, and azimuth/elevation AoAs and AoDs follow the uniform distribution $\mathcal{U}[-\pi/2, \pi/2]$ with angle spread of $7.5^{\circ}$. Within each cluster, there are $N_{\rm ray}=10$ rays. As for the antennas, we consider transmitter adopt $8\times8$ UPA with hybrid precoder, the receiver adopt $2\times2$ UPA with fully-digital combiner, and the distance between each adjacent antennas is half wavelength. Moreover, we consider the number of RF chains at transmitter is $N_t^{\rm RF}=4$ and the data stream is $N_s=3$. Additionally, we consider 4 types of classical FS patterns shown in Fig. \ref{FS}, where antenna elements with the same color share the same RF chain.
\begin{figure}[t]
	\centering
	\includegraphics[width=1\columnwidth, keepaspectratio]{fixed.eps}
	\caption{Four types of FS: (a) Vertical type; (b) Horizontal type; (c) Squared type; (d) Interlaced type.}\label{FS}
\end{figure}

Throughout this part, following baselines will be considered
for performance benchmarks: \textbf{Optimal fully-digital} scheme considers the fully-digital MIMO system, where the SVD-based precoder/combiner is adopted as the performance upper bound. \textbf{Simultaneous OMP (SOMP)} scheme is an extension version of the narrow-band OMP-based spatially sparse precoding in \cite{OMP}. In broadband, SOMP-based hybrid precoding scheme can simultaneously design the RF precoder/combiner for all subcarriers. \textbf{Discrete Fourier transform (DFT) codebook} scheme designs the RF precoder/combiner from the DFT codebook instead of steering vectors codebook in SOMP scheme \cite{mao}. \textbf{Covariance eigenvalue decomposition (EVD)} scheme estimates the covariance matrix of the channels using the mean of auto-correlation matrices at each subcarrier \cite{170825}. The RF precoder is designed based on the EVD of the covariance matrix of the channels.

\begin{figure}[tb]
    \centering \subfigure{\includegraphics[width=.49\columnwidth, keepaspectratio]{seprefull.eps}}
    \subfigure{\includegraphics[width=.49\columnwidth, keepaspectratio]{sepresub.eps}}\\
	\caption{SE performance comparison of different hybrid precoder schemes: (a) FCA with $Q=\infty$ and $Q=3$; (b) PCS with $Q=\infty$.}\label{fig_pre}
\end{figure}
In Fig. \ref{fig_pre}, we compare the SE performance of the proposed hybrid precoding scheme with the baselines, where both FCA and PCS are investigated. In Fig. \ref{fig_pre} (a), for FCA, our proposed PCA-based hybrid precoding scheme outperforms conventional DFT codebook-based hybrid precoding scheme and SOMP-based hybrid precoding scheme. Both the proposed PCA-based hybrid precoding scheme and covariance EVD-based hybrid precoding scheme have the very similar performance, and they suffer from negligible performance loss when compared to the optimal fully-digital scheme. This is because mmWave MIMO channels associated with different subcarriers share the same row space due to the common scatterers. Meanwhile, our proposed algorithm can exploit the principal components of the common row space to establish the hybrid precoder. The SOMP-based and DFT codebook-based hybrid precoding schemes work poorly, since their analog codebooks are limited to the steering vector forms. Finally, it can also be observed that the influence of quantization in phase shifters is negligible for our scheme. As for the PCS, Fig. \ref{fig_pre} (b) shows that our scheme outperforms conventional covariance EVD-based hybrid precoding scheme with different FS patterns and DS. The antenna grouping scheme in \cite{170825} considers a greedy approach, which may lead to the imbalance antenna grouping by acquiring local optimal solution. By contrast, the shared-AHC algorithm for DS grouping introduces the mutually correlation metric (\ref{mul_cor}), which can efficiently avoid this issue. Therefore, the proposed shared-AHC algorithm for DS grouping outperforms it counterpart in \cite{170825}.

\begin{figure}[tb]
    \centering \subfigure{\includegraphics[width=.49\columnwidth, keepaspectratio]{eepassive.eps}}
    \subfigure{\includegraphics[width=.49\columnwidth, keepaspectratio]{eeactive.eps}}\\
	\caption{EE performance comparison of different hybrid precoding schemes on different antenna architectures: (a) Passive antenna; (b) Active antenna with.}\label{fig_EE}
\end{figure}
In Fig. \ref{fig_EE}, we compare the EE performance of the proposed PCA-based hybrid precoding scheme and the baselines with FCA and PCS, where both passive and active antenna architectures are investigated. Note that the power values of key electronic components can refer to Section V.
In Fig. \ref{fig_EE} (a), for passive antenna architecture, the EE performance of PCS by using the proposed PCA-based hybrid precoding scheme outperforms that of FCA by using the SOMP-based and DFT codebook-based hybrid precoding schemes. The reason is that PCS adopts a much smaller number of phase shifters than FCA. Moreover, DS outperforms the other FS patterns in SE, and it consumes the same power with the other FS patterns. Therefore, DS outperforms other four types of FS patterns. It is worth mentioning that the optimal fully-digital scheme has the worst EE performance, since the numbers of power-consuming PAs, DACs, and mixers are proportional to that of antennas.
In Fig. \ref{fig_EE} (b), for active antenna architecture, the advantage of EE performance for different FS patterns by using the proposed hybrid precoding scheme over the FCA with several typical hybrid precoding schemes and optimal fully-digital precoding scheme is not considerable. This is because active antenna architecture requires the power-hungry PAs for each antenna. Meanwhile, the advantage of the reduced power consumption of FS structure is greatly weakened by its disadvantage in SE performance when compared to FCA. Finally, the EE performance of DS with the proposed hybrid precoding scheme still has the obvious advantage over the baselines and four typical types of FS with the proposed scheme. This reveals the appealing advantage of DS in practical situation when both the power consumption and SE should be well balanced.

\section{Conclusions}
This paper has proposed a hybrid precoding scheme based on machine learning for broadband mmWave MIMO
systems with DS. We first acquire the low-dimensional frequency flat precoder from the optimal frequency-selective precoders based on PCA for FS. Then, we extend the proposed PCA-based hybrid precoder design to the DS. We propose the shared-AHC algorithm inspired by cluster analysis in machine learning for antenna grouping to further improve the SE performance. Additionally, we analyze the EE performance for FCA, FS, and DS with passive and active antennas. Simulations further verify the proposed PCA-based hybrid precoding scheme has the better SE and EE performance than conventional schemes.

\section*{Acknowledgment}
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61471037, 61701027, and 61201181), the Beijing Natural Science Foundation (Grant No. 4182055), Huawei Innovation Research Program (HIRP), and Youth Project of China Academy of Information and Communications Technology.

redpajama_set_name: RedPajamaArXiv


Revelation BibleCourse Lesson 38 from Jerry L. Wells on Vimeo.
1. Where are the angels to pour out their vials?
Revelation 16:1 "And I heard a great voice out of the temple saying to the seven angels, Go your ways, and pour out the vials of the wrath of God upon the earth."
Answer upon the earth to question 1.
2. Where did the first angel pour out his vial?
Revelation 16:2 "And the first went, and poured out his vial upon the earth; and there fell a noisome and grievous sore upon the men which had the mark of the beast, and upon them which worshipped his image."
Answer earth to question 2.
3. Did all the people receive a noisome and grievous sore?
4. Which people received the sores? Choose one of the following.
A. Those who had the mark of the beast.
B. Those who had the number of the beast.
C. Those who worship the image of the beast.
Answer E to question 4.
5. Where did the second angel pour out his vial?
Revelation 16:3 "And the second angel poured out his vial upon the sea; and it became as the blood of a dead man: and every living soul died in the sea."
Answer upon the sea to question 5.
6. The became as the of a dead man.
Answer sea blood to question 6.
7. Where did the third angel pour out his vial? .
Revelation 16:4 "And the third angel poured out his vial upon the rivers and fountains of waters; and they became blood."
Answer rivers and fountains to question 7.
8. What did the water become?
9. Who praised God when the third angel poured out his vial?
Revelation 16:5 "And I heard the angel of the waters say, Thou art righteous, O Lord, which art, and wast, and shalt be, because thou hast judged thus."
Answer angel of the waters to question 9.


I cannot explain how excited and nervous I am to be doing this next interview. I've been hoping to meet this guy for so long that now I'm finally doing it, I'm so nervous I'm sure I'll forget what to ask him.
Ladies and gentlemen, please welcome the intergalactic movie phenomenon, Jake Elloway.
Hi Jake, I'm so happy to meet you. Thank you so much for being here, I know you're a busy guy.
Hi there, I'm delighted to be here.
The story of your time in space has just come out. That must've been a scary experience for you?
Oh yeah it was, very scary at first but y'know, it's amazing how quickly you adjust in a crisis. I would never have dreamed of anything like that happening to me, although being a celebrity has its dangers of course, and when something like that happens, you just find something inside yourself to help you cope. I think your mind takes over and blocks out a lot of the emotion so that you're operating in auto pilot and this helps you through the worst of the crisis moments.
The people you met on board the space ship weren't all happy to see you were they? Was that a surprise?
Not exactly a surprise no. There will always be a percentage of people who don't get what I do or don't like it and that's okay. You can't please everyone. What did surprise me was the speed at which they were prepared to give me up. I like to think I'm a nice guy but some of those guys would've happily handed me over without a second thought. It took some time for me to win them over and I only achieved that through showing them that I was capable of being just an ordinary guy and working hard and pulling my weight. By the time I left several of them had spoken to me and apologised for being off with me when I first came aboard and that touched me.
You umm, met some ladies aboard the ship didn't you?
What was it actually like down on Tellizon 4? How did you cope with all those awful creatures?
I still ask myself that question sometimes, when I think back to it. It's one of those situations that's so out of your sphere of imagination that you just can't prepare yourself for it. If I'd known in advance it was going to happen, I'm pretty sure I'd have been a total mess and probably wouldn't have survived more than a few minutes. Because it just happened without time to prepare, I just had to get on with it and something just sort of, took over. Call it instinct or a higher power or whatever but whatever it was, I'm grateful. So back to you question, what was it like? Well, it was hot and humid like you wouldn't believe. We spent our days sweating and our nights shivering and the creatures were just, well they were umm, they were something else. I almost got killed so many times by huge things and tiny things, cute looking things and ugly things. You just can't trust anything down there. I have a bit of a fear of bugs anyway and the ones on Tellizon 4 were horrific. I had nightmares about it for a while and it made me more scared of the bugs back home than I usually was. It actually began to interfere with my career so my staff arranged for some counselling for me and within a few weeks I had everything back into a healthy perspective again.
You made a few good friends though didn't you? Do you keep in touch with them?
Oh yeah those five guys are real special to me. Nef now works with me as my Personal Martial Arts Trainer and it's great to have him around. He's such a nice guy and seems to know the real me inside without me having to make an effort. He has this ability to see through the facade that I have to put on most of the time as a celebrity and he's not phased by my celebrity status as all. He's my best buddy and he always knows when something's bothering me and always manages to get me to talk about it, even when I don't want to. I keep in touch with the others regularly and they come and visit when they're on leave. I've had them all stay with me a few times since I left the ship and Nef and I have been invited to Titch and Rosa's wedding next year. And I even got to meet Sam again.
Sam Sinclair? The law enforcement guy?
Yeah, didn't you know he was here with me today?
He's here with you? Really?
Yeah, *laughs at my incredulity* sure he's here. Hey Sam, *turns around and yells at the door, which opens and in bounces Sam Sinclair* come on in here buddy.
Wow, how awesome to have you both here together. Have you actually met since it all happened?
Jake – yeah we met once about umm, three months or so ago wasn't it? *looks at Sam who nods* when he visited Earth on a job.
Sam – It's great to see you again hunny and yeah it was about three months ago. I can honestly say that I've never slept in such a swanky place in my whole life.
Jake – This guy here is a real hero. Don't let him brush off his skills and tell you it's nothing. He does an awesome job and seldom gets the thanks he deserves.
Jake – Well I'm off on another space trip soon to make a movie. This one means six months away from Earth but this time I'm taking my family and key staff with me. My mother would never forgive me for leaving for six whole months. Nef is coming too and I'm confident that we won't have any problems like the last time I was in space.
It's about a group of soldiers who have a weird experience while helping set up a new military base on a moon somewhere.
And Sam, what are you up to lately?
Well I'm actually here to shop for a new ship. I lost my old gal a few weeks ago during a particularly difficult job and lost everything. Luckily I didn't lose much personal stuff but it's still a nuisance. Once I'm mobile and restocked I'm right back to work again. I'm working on uploading another video log to the galactic web so you'll get to find out about my time on Deligon 2. It's actually interesting Jake because in that job, I lost my best buddy and he was a Damiklonian too.
Well good luck with the movie Jake, or should I say break a leg?
And good luck with your ship hunting Sam and I look forward to your next story.
Thank you, I'm sure I'll find the right one soon.
Thank you so much, both of you. It's been the most exciting time meeting you both.
Sam – It's a pleasure, thank you. I'll invite myself back when my next story comes out.


text:
← Celebrate Medicare with Alachua County Labor Coalition
In memory of Harriet Ludwig, activist, reporter →
GRACE celebrates four years of ending homelessness
By Jon DeCarmine, Director, GRACE
When I think of what we've been able to accomplish at GRACE in the past four years, I think of a poster I have up on the wall in my office. It's a quote from St. Francis of Assisi, and it reads: "Start by doing what's necessary; then do what's possible; and suddenly you are doing the impossible."
For years, people without houses had to spend all day, every day, navigating a frustrating maze of appointments and expectations at dozens of different agencies spread out all over town. All the while, they had to be thinking about where would sleep that night, where they would use the bathroom, and where they could store their belongings. It was a lot to ask, and it left a shockingly high percentage of our homeless population out in the heat and in the cold, unsure what to do or how to get it done.
Fast forward to where we are with GRACE today — we have a one-stop assistance center and emergency shelter designed to eliminate all of the barriers that kept people from getting the help they need. Our services are in one location, and we have a team of trained professionals ready to help from the moment someone shows up. We serve the people who can't or won't go to other shelters in town, and we do it on a scale no one thought was even possible.
Our community, finally, has everything it needs in an emergency shelter. People have a place to go where they can get the services they need to get off the street. Our guests are safe — we've had no major violent incidents since the day we opened. People have access to food, to shelter, to clothing, bathrooms, showers, laundry, and a wide variety of partner agencies with a common goal of getting them off the streets as quickly as possible. People have shelter from the cold, and a safe place to go during hurricanes and tropical storms. Our Housing Programs regularly move our most vulnerable citizens off the streets and into Rapid Rehousing and Permanent Supportive Housing apartments.
In four short years, we've grown from a glimmer in the eye of a small, committed group of people to a 113-bed shelter using best practices and data to reduce homelessness in Alachua County. There are hundreds and hundreds of people in homes – right now, thanks to support they received at GRACE – who were homeless five years ago. Homelessness in our community is down 30 percent. We've done what's necessary, and we've done what's possible. Suddenly, we're doing the impossible. And we're just getting started.
Looking for a way to help? We need summertime volunteers in the Cafe to help prepare and serve meals. Cooking at Cafe 131 is a great way to get together with friends from school or work, and you'll be making an immediate impact. Our volunteers have helped serve almost half a million meals since 2014, and we'd love to see you out here this summer!
To volunteer as an individual or as part of a meal group, please contact Travis at 352-792-0800 ext 104, or e-mail him at volunteer@gracemarketplace.org. D

redpajama_set_name: RedPajamaCommonCrawl


text:
Watts thanks Samuel L. Jackson for 'comfortable' post-natal sex scenes
By Celebrity Buzz	on April 30, 2010 at 6:04 PM
Actress Naomi Watts has heaped praise on her Mother & Child co-star Samuel L. Jackson for making nude sex scenes just eight weeks after she gave birth as pleasant as possible. The couple literally …
Mackenzie Phillips drug case dropped
A felony drug charge against Mackenzie Phillips has been dropped after the embattled actress successfully completed a drug rehabilitation program. Phillips was arrested for cocaine and marijuana …
Badu pleads not guilty to disorderly conduct charge
Erykah Badu has pleaded not guilty to a charge of disorderly conduct after a female bystander made an official complaint about the singer's public nudity during a controversial video shoot. The star …
Boyle's debut is best selling album in the world
Susan Boyle's debut album has been named the world's best-selling record of 2009. I Dreamed A Dream, which hit stores in November 2009, landed the title of Britain's top-selling release last year, …
Ricky Martin: Arizona immigration law 'makes no sense'
APThe appearance at the 2010 Billboard Latin Music awards was Ricky Martin's first since his admission that he was a "fortunate homosexual man". Singer Ricky Martin joined in the criticism of …
Letterman attempts to put life back together after sex scandal
Associated PressThis screen-grab shows David Letterman as he tells his audience that he had sexual relationships with female employees. David Letterman says the scandal surrounding his workplace …
James Cameron to help build 3D camera for Mars rover
By TracyS	on April 30, 2010 at 12:17 PM
APJames Cameron's Avatar is currently the highest-grossing film in history. "Avatar" director James Cameron is helping to build a high-resolution 3-D camera on the next-generation Mars rover. The …
Swiss officials reject secret testimony in Polanski case
By Celebrity Buzz	on April 30, 2010 at 12:03 PM
In pictures: Polanski's troubled life • Roman Polanski timeline Swiss justice officials on Friday rejected an attempt by Roman Polanski's U.S. lawyers to unseal secret testimony by his original …
Bret Michaels may be ready to rock within weeks
By Celebrity Buzz	on April 30, 2010 at 11:02 AM
APBret Michaels is best known as the lead singer of glam rock band Poison. Rocker Bret Michaels could stage yet another comeback sooner than expected after he suffered a brain hemorrhage last week. …
Chris Brown curses radio DJ, complains they don't play his songs (w/audio)
Chris Brown upset fans in Richmond, Virginia on April 30 by aiming a foul-mouthed rant at a radio presenter and walking out of an interview. The singer has been attempting to rebuild his career after …


Hi, I am an Indian national married to a British citizen. Anyone who is applying for a visa will know how important it is to find the right lawyer. I met 2 different lawyers before meeting Tito and within 15 min of our meeting, I was convinced that he will be the one to represent my case. He is very professional and has in-depth knowledge of immigration law. He is very genuine and I recommend him to everyone. Great job Tito, you are the best!


text:
@interface SCTMasterViewController : UIViewController

@end

redpajama_set_name: RedPajamaGithub


Generic drug Amoxapine is considered just as safe and effective as its brand-name equivalents such as Asendin. Did you know that buying the generic drug Amoxapine from IDM is much cheaper than buying the Asendin brand drug?
Amoxapine is an antidepressant used in the treatment of depression, agitation, anxiety disorders, bipolar disorders, and panic disorders. It belongs to a group of drugs called tricyclic antidepressants. Amoxapine works by restoring the balance of natural chemicals in the brain. This medicine may also be used to treat other conditions as determined by your doctor.
What is the recommended dosage of Amoxapine?
The dosage of Amoxapine prescribed to each patient will vary. Always follow your physician's instructions and/or the directions on the prescription drug label.
Amoxapine can be taken with or without food.
What if you miss a dose of Amoxapine?
If your physician has instructed or directed you to take Amoxapine medication in a regular schedule and you have missed a dose of this medicine, take it as soon as you remember. However, if it is almost time for your next dose, then skip the missed dose and go back to your regular dosing schedule. Do not double the doses unless otherwise directed.
What if you overdose on Amoxapine?
Any medication taken in excess can have serious consequences. If you suspect an overdose of Amoxapine, seek medical attention immediately.
What other drugs could interact with Amoxapine?
It may be noted that drugs other than those listed above may also interact with Amoxapine.
Usually drug interactions occur when it is taken with another drug or with food. Before you take a medication for a particular ailment, you should inform the health expert about intake of any other medications including non-prescription medications, over-the-counter medicines that may increase the effect of Amoxapine, and dietary supplements like vitamins, minerals and herbal, so that the doctor can warn you of any possible drug interactions.
Amoxapine can interact with MAO inhibitors.
What are the side effects of Amoxapine?
Like other medicines, Amoxapine can cause some side effects. If they do occur, the side effects of Amoxapine are most likely to be minor and temporary. However, some may be serious and may require the individual to inform the doctor or visit the nearest hospital immediately.
It is pertinent to note that side effects of Amoxapine cannot be anticipated. If any side effects of Amoxapine develop or change in intensity, the doctor should be informed as soon as possible.
Amoxapine can cause side effects such as drowsiness, dizziness, constipation, dry mouth, headache, fatigue, and insomnia. This is not a complete list of all side effects. Do concur with your doctor and follow his directions completely when you are taking Amoxapine.
What are the questions to ask your doctor before taking Amoxapine?
Is it possible for me to take Amoxapine with other drugs?
Should certain beverages, foods and other products be avoided when I take Amoxapine?
What are the possible drug interactions of Amoxapine?
How will Amoxapine work in my body?
How should Amoxapine be taken?
How to reduce the risk of Amoxapine drug interactions and side effects?
The health and medical information provided here is intended to supplement and not substitute for the expertise and judgment of your physician, pharmacists or other health care professional. It should not be understood to indicate that the use of Amoxapine is safe, appropriate or effective for you. Always consult your health care professional before using this, or any other, drug.


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We are looking for a strong Manager of Business Development who is excited by the thought of building businesses with online content Creators. This role will be responsible for supporting the VP of Creator Partnerships with all projects on a global scale. You will work to bring in Creator Joint Ventures (JVs) from lead generation to close and beyond.
Please include a resume and cover letter.
We receive a lot of applications, but we will notice a well-written and fun letter that shows us why you are passionate about Creator first businesses.
Next 10 Ventures is an operating and investment company focused on incubating and accelerating new businesses, content, products and services within the global Creator economy. Our mission is to enrich, inspire, and entertain kids and young adults via the investments and partnerships we establish with Creators who thrive as artists, entrepreneurs, educators and opinion leaders.


'''''myservice.call_method.post''''' would be generated for a call to POST '''''<nowiki>http://example.com/api/v2/myservice/call_method</nowiki>'''''.
The core thing is to define prefixes for namespaces and use them to fortify each and every tag.
It's much better to use default types for fields whenever you can - so for password you have to list the type, for the Nonce it's already Base64.
Make sure that you check that the generated token is correct before you send it via XML and don't forget that the content of wsse:Password is Base64( SHA-1 (nonce + created + password) ) and date-time in wsu:Created can easily mess you up. Nonce can have different encodings so if you really want to force another encoding you'll have to look further into wsu namespace.
The core thing is to define prefixes for namespaces and use them to fortify each and every tag. It's much better to use default types for fields whenever you can - so for password you have to list the type, for the Nonce it's already Base64.


One of the most important benefits of placing a dining table outdoors is that it provides the whole place with joy and entertainment during having summer dinners out. The country yard is definitely an expansion of the living room where you have your space doubled so you will be able to be with your family and friends over almost any meal. There are many advantages of having a dinner table like amusing your guests and invite them to have a luxurious barbecues dinner and spend great time during summer afternoons. Placing a dinner table along with some chairs outside will guarantee you a true outdoor dining experience.
 Wood Outdoor Dining Tables. Believe it or not, one of the most desired materials to make dining room tables is wood. They are available in countless designs for several purposes like picnic benches, tables designed for two persons or even round tables. If you want something designed especially for the outdoors, wicker dining sets or dark wood tabletops are completely what you need.
 Stone Outdoor Dining Tables. Dining tables made of stone has many buyers to buy due to its ability to resist the nature elements. Therefore, you offer the best as long as you can. You are free to go for the completely stone set or you can only go for the tabletop, which is also made of stone. All you need to do in terms of maintenance is to wash the tables frequently from time to time to keep it clean.
 Glass-top Dining Tables. They are very appropriate for covering decks and patios. Glass-top tables improve how your dining area looks. However, they are weak and delicate. If you intend to buy a glass-top table a higher value, you should go for a metal-framed glass dining table. If you are working on improving the way your garden looks, you can opt for weatherproof wooden frames.
Finally, it is time to mention some basic essential you are going to need to consider before buying an outdoor dining table and during placing it. Make sure that every simple thing you are going to put outdoors is weatherproof. Nothing better than buying an outstanding quality of anything especially when it comes to outdoor furniture because the better quality you choose, the more it lasts. Take into account the quality of lighting as well to get rid of bugs.


text:
She could have found out my first name, of course -- that wouldn't be difficult.
Page "romance" ¶ 748
from Brown Corpus
She and could
She showed her surprise by tightening the reins and moving the gelding around so that she could get a better look at his face.
She remembered little of her previous journey there with Grace, and she could but hope that her dedication to her mission would enable her to accomplish it.
She could not scream, for even if a sound could take shape within her parched mouth, who would hear, who would listen??
She wished that she could talk to her mother about it.
She realized I'd have to notify the police, but fervently hoped I could avoid mentioning her name.
She even devised a system of colors, whereby the boy could easily distinguish the different note values.
She could not resist the opportunity `` of showing her superiority in argument over a man '' which she had remarked as one of the `` feminine follies '' of Sara Sullam ; ;
She read everything else she could get her hands on, including an article ( she thinks it was in the Atlantic Monthly ) by Mark Twain on `` White Slavery ''.
She could not face coffee or tea without milk, and was always craving types of food that were not available aboard a sailing ship.
She ended her letter with the assurance that she considered his friendship for her daughter and herself to be an honor, from which she could not part `` without still more pain ''.
She was wise enough to realize a man could be good company even if he did weigh too much and didn't own the mint.
She could act and she could write.
She was not a girl, he could see.
She thought again of her children, those two who had died young, before the later science which might have saved them could attach even a label to their separate malignancies.
She felt the look and looked back because she could not help it, seeing that he was neither as old nor as thick as she had at first believed.
She also taught them to sing `` I wish I could shimmy like my sister Kate ''.
She could see that Mr. Gorboduc was intrigued ; ;
She found she could cope with all kinds of problems for which she was once considered too helpless.
She thought she had great possibilities in the ballet and wanted to show the eminent producer how well she could dance.
She could do no wrong at the tables that time.
She didn't like her stepmother, but nothing is known to have occurred shortly before the crime that could have caused such a murderous rage.
) She snarled terribly but intuition told him, again, that she was bluffing, and he could see that half her attention was distracted by the dogs.
She whirled and faced him, roaring terribly, and Ulyate, watching through the leaves, could not understand why she did not charge and obliterate him, because he wouldn't have much of a chance of getting away, in that thick growth, but she seemed just a trace uncertain ; ;
She and have
She seemed to have come such a long distance -- too far for her destination which had wilfully been swallowed up in the greedy gloom of the trees.
She didn't have the heart.
She might have been someone he had once loved.
She wrote in her journal, `` I have not heard the least profane language since I have been on board the vessel.
She used to tell me, `` When I stand there and look at the flag blowing this way and that way, I have the wonderful, safe feeling that Americans are protected no matter which way the wind blows ''.
She was the opposite of everything she should have been -- a positive pole in a negative home, a living reaction of warmth and kindness to the harsh reality of her father.
`` She wants you and Barbara to have dinner with her tomorrow night ''.
She usually wore weeds, and a stranger watching her board a train might have guessed that Mr. Pastern was dead, but Mr. Pastern was far from dead.
She would have said triumph.
She had surprised Hans like she had surprised me when she said she'd go, and then she surprised him again when she came back so quick like she must have, because when I came in with the snow she was there with a bottle with three white feathers on its label and Hans was holding it angrily by the throat.
She ascribed her delight with both experiences to the effect they seemed to have of temporarily removing from her the controls which she felt so compulsively necessary to maintain even when it might seem appropriate to relax these controls.
She teamed up with another beauty, whose name has been lost to history, and commenced with some fiddling that would have made Nero envious.
She later divorced Graham, who is believed to have moved to Bolivia.
She must have looked temptingly pretty to the dean as he put the crown on her head.
She would have been taking more than a fair risk of being seen and recognized during her travels.
She was closing and within one more bound would have been able to reach the rear end of the bay, but -- and here Jones and Loveless and Ulyate were holding breath for all they were worth -- she never quite caught up that last bound.
She might have been talking to some of her friends about her husband if they've been having any trouble ''.
She refused to have a doctor, insisting there was nothing a doctor could do for her.
She might, conceivably, have brought one in in a large-enough suitcase.