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Industry shares PHMSA's pipeline safety goal of zero incidents
WASHINGTON, October 1, 2015 – API Midstream Group Director Robin Rorick said industry shares the goal of zero incidents that is intended by the new hazardous liquids pipeline proposed rule by the Pipeline and Hazardous Materials Safety Administration (PHMSA).
Safe offshore energy development remains key driver for economic development and energy security
WASHINGTON, September 14, 2015 – Ahead of tomorrow's congressional hearing in New Orleans, API highlighted the importance of offshore oil and natural gas development to the region's economic development.
API: Industry embraces new pipeline safety framework
WASHINGTON, July 8, 2015 – The American Petroleum Institute announced a new pipeline safety management system standard. This new standard which was built on the industry goal of zero incidents was created with engagement and guidance from the National Transportation Safety Board (NTSB), the Pipeline and Hazardous Materials Safety Administration (PHMSA), and other key stakeholders to further enhance pipeline safety.
API: Offshore oil & gas operations are safer than ever
WASHINGTON, April 9, 2015 – Advances in technology, standards and practices in the last five years have made America's offshore oil and natural gas industry safer than ever, API President and CEO Jack Gerard told reporters on a press conference call today.
© Copyright 2015, API Energy


text:
West Virginia State Penitentiary – Part of the Top Ten Most Violent Correctional Facilities list
Designed in a Gothic style and located in Moundsville, West Virginia, this prison served its time well until the day when it was retired from use in 1995. The design itself bears a striking resemblance with the 1858 state prison in Joliet, Illinois.
Both of the structures are almost completely identical complete with turrets and battlements. The only difference is the West Virginia State Penitentiary is half the size.
The West Virginia State Penitentiary.
During 1863, West Virginia withdraws from Virginia at the height of the American Civil War. As a result, this new state had a shortage of various public institutions, among which a prison. And so the state built a temporary wooden prison to help satisfy the momentarily needs of the state and also allowed prison officials time to consider the right design to be used.
The first building to be built was the North Wagon Gate. It was constructed with hand-cut sandstone that was quarried from a local site. As was the case with many other prison constructions, the state of West Virginia used prison labor during the construction.
View from a hallway into a cell.
This was the first phase of the construction that continued until 1876. Once it was done, the total costs were around the $360,000 range. After the North Wagon Gate, next to follow were the north and south cell block areas.
A portion of a cell block at the West Virginia State Penitentiary.
The South Hall had 224 cells, and the North Hall had the kitchen, dining area, hospital, and chapel. In between the two, there was a 4-story tower that connected the South and North Halls and served as the administration building.
Inside this building, there was a space for the female prisoners as well as the personal living quarters for the warden and his family. The prison then opened its gates to receive 250 inmates before the next phase of building began on prison workshops and other secondary facilities.
A portion of a cell block.
During the 1900s some of the industries that could have been found inside this prison included a carpentry shop, a paint shop, a workshop, a stone yard, a brickyard, a blacksmith, a tailor, a bakery, and a hospital. At a later date, there were even a school and a library added to the arsenal of buildings the prison had.
The former dentist's office.
When all was put together, the prison was virtually self-sufficient. There was also a prison farm and a coal mine that helped with the electricity, reducing the electricity bill by as much as $14,000 a year.
According to the Warden's report "both the quantity and the quality of all the purchases of material, food, and clothing have been very gradually, but steadily, improved, while the discipline has become more nearly perfect and the exaction of labor less stringent."
But as time went on the conditions at the prison started to worsen so much so that this facility was ranked by the United States Department of Justice's Top Ten Most Violent Correctional Facilities list. From 1899 to 1959, ninety-four men were executed at the prison.
The inmates' recreation room which they called "The Sugar Shack" was renowned as a place where gambling, fighting, and raping was done. With time the prison started to get overcrowded, often a cell 5 feet by 7 feet was shared between three prisoners. When the decision to expand the facility came in 1929, construction was once more accomplished using prison labor.
Urinal and crude pipe fixtures.
But none the less, it continued to be more than challenging to keep the prisoners in order. There was a total of thirty-six homicides that took place in the prison. Toward the end, it became well known for riots and escapes. In one prison break on November 7th, 1979, fifteen inmates escaped. A few years later in January of 1986, what was considered to be the most infamous riot in recent history took place.
The fate of the West Virginia State Penitentiary was sealed in a 1986 ruling by the West Virginia Supreme Court to close down the facility. It is now run as a tourist attraction that offers prison tours and events for ghost hunters.
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redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
\label{Introduction}
The deployment of large numbers of antennas at a cellular
base station (BS) to communicate with multiple user terminals
has received a considerable amount of
attention recently \cite{RUSEK,LARSSON}. Specifically, large (a.k.a. massive)
multiuser multiple-input multiple-output (MU-MIMO) systems have been
shown to achieve orders of magnitude greater performance than conventional
MU-MIMO systems, due to their ability to leverage
favorable propagation conditions \cite{LARSSON}.
Nevertheless, the emergence of such
systems has posed new engineering challenges which must be overcome
before their adoption on a scale commensurate with their true potential.
One of the critical issues is accommodating large numbers of antennas in
fixed physical spacings \cite{MASOUROS1,MASOUROS2}. This tends to
increase the level of spatial correlation and antenna coupling, as
successive elements are placed in close proximity with inter-element
spacings less than the desired half-a-wavelength \cite{MASOUROS1}.
This is known to cause a detrimental impact on the terminal
signal-to-interference-plus-noise-ratio (SINR) and system spectral
efficiency. It is thus important to rigorously
analyze and evaluate the performance of systems with space-constrained (SC)
antenna arrays.

Numerous works have investigated the impact of SC
antenna arrays on the performance of large MU-MIMO systems (see
e.g., \cite{MASOUROS1,MASOUROS2,NGO,GARCIA-ROD,ZHANG,BISWAS,GE} and
references therein). Specifically, \cite{MASOUROS1} analyzed the
ergodic sum spectral efficiency of large MU-MIMO systems with
fixed array dimensions. The authors in \cite{MASOUROS2} demonstrated that
multiuser interference does not vanish in SC MU-MIMO systems with
growing numbers of antennas. The uplink performance
with maximum-ratio combining (MRC), zero-forcing and
minimum-mean-squared-error receivers has been analyzed in
\cite{NGO,ZHANG} where the authors derive upper and lower bounds on the
ergodic sum spectral efficiency. Moreover,
\cite{GARCIA-ROD,BISWAS,GE} investigated the
energy efficiency performance of SC systems with various
large-scale antenna array topologies considering antenna coupling.

However, very few of the above mentioned studies\footnote{We make an
exception in \cite{MASOUROS2}, which considers
pure LoS channels. This is an extreme case, which in general
may not be realizable in practice, even at mmWave frequencies, where on
average 1-3 scattering clusters are anticipated in the
propagation channel (see e.g., \cite{AKDNENIZ}).} consider the effects of
line-of-sight (LoS) components, which may be a dominant
feature in future wireless access with the use of smaller cell sizes, potentially
operating in the millimeter-wave (mmWave) frequency bands
\cite{SUN,TATARIA,TATARIA2}. Hence,
understanding the performance of SC systems with LoS presence,
i.e., with Ricean fading
is of particular importance.
Moreover, the respective channel models in \cite{NGO,ZHANG,GE}
assume that all terminals are seen by the BS array via the same set of
incident directions, resulting in common (equal) spatial correlation structures.
In reality, differences in the local scattering around the physical location of
each terminal gives rise to wide variations in the correlation patterns \cite{NAM}.
In addition to the small inter-element spacings,
this further contributes to the level of correlation in the channel,
impacting the terminal SINR and system spectral efficiency. Thus,
to more accurately capture the correlation differences in
multiple channels, we consider distinct correlation matrices for
each terminal.
Motivated by the aforementioned considerations, with a SC
uniform linear array (ULA), we present a
framework for analyzing the expected per-terminal
SINR and ergodic sum spectral efficiency of large MU-MIMO
systems with MRC at the BS.
Specifically, our main contributions are as follows:
\begin{itemize}
 \item We analyze the performance of MU-MIMO systems with
 SC ULAs under correlated Ricean
 fading channels. In doing so, we extend and generalize the
 SC channel models presented in
 \cite{MASOUROS1,MASOUROS2,NGO,GARCIA-ROD,ZHANG,BISWAS,GE}
 to cater for unequal correlation matrices and
 unequal Rice factors for each terminal.
 To the best of the authors' knowledge, such generality in the
 channel model has not previously been considered.
 \item With MRC at the BS, we derive tight closed-form
 approximations to the expected per-terminal SINR and
 ergodic sum spectral efficiency. We show that a SC
 antenna deployment causes a saturation of the expected SINR
 with increasing numbers of BS antennas and operating
 signal-to-noise-ratios (SNRs).
 \item With a fixed number of terminals, as the number of BS antennas
 increases without bound, we derive novel limiting expected
 SINR and ergodic spectral efficiency
 expressions to demonstrate the convergence behavior of large
 SC MU-MIMO systems.
 \item Finally, we present special cases of the
 derived analytical results when NLoS components are present with
 equal and unequal correlation matrices, as well as, when
 each terminal having LoS has fixed correlation matrices.
\end{itemize}

\textbf{Notation.} Boldface upper and lower case symbols denote matrices
and vectors, respectively. Moreover,
$\bm{I}_{M}$ denotes the $M\times{}M$ identity
matrix. $\left(\cdot\right)^{\textrm{T}}$,
$\left(\cdot\right)^{\textrm{H}}$ and $\left(\cdot\right)^{-1}$ denote the
transpose, Hermitian transpose and inverse operators, respectively. We use
$\left[\bm{H}\right]_{i,j}$ to refer to the $\left(i,j\right)$-th element of
$\bm{H}$, whilst $\bm{h}\sim\mathcal{CN}\left(\mu,\sigma^{2}\right)$ denotes
a complex Gaussian distribution for $\bm{h}$, where each element of $\bm{h}$
has a mean $\mu$ and variance $\sigma^{2}$.
We use $x\sim{}u\left[a,b\right]$ to denote a uniform random variable for
$x$ taking on values from $a$ to $b$. $||\cdot{}||$,
$||\cdot{}||_{\textrm{F}}$ and $|\cdot{}|$ denote the standard two norm,
Frobenius norm and scalar norm, respectively. Finally,
$\textrm{tr}\left[\cdot\right]$ and $\mathbb{E}\left[\cdot\right]$ denote
the trace and statistical expectation operations.

\vspace{-4pt}
\section{System Model}
\label{SystemModel}
\vspace{-1pt}
We consider the uplink of a large MU-MIMO system operating
in an urban microcellular (UMi) environment, where $L$ non-cooperative
single-antenna user terminals transmit data to $M$ receive antennas at the BS
$\left(M\gg{}L\right)$ in the same time-frequency interval. The BS comprises
of a ULA with equispaced, omnidirectional antennas.
We assume channel knowledge at the BS with narrow-band transmission and
no uplink power control. The composite $M\times{}1$
received signal at the BS array can be written as
\vspace{-3pt}
\begin{equation}
 \label{receivedsignal}
 \bm{y}=\rho^{\frac{1}{2}}\bm{G}\bm{D}^{\frac{1}{2}}\bm{s}+\bm{n},
 \vspace{-4pt}
\end{equation}
where $\rho$ is the average transmit power of each terminal, $\bm{G}$ denotes
the $M\times{}L$ fast-fading uplink channel matrix between $M$ BS antennas
and $L$ terminals (discussed further in Section~\ref{ChannelModel}),
$\bm{D}$ is an $L\times{}L$ diagonal matrix of link gains for the $L$ terminals
in the system, such that $\left[\bm{D}\right]_{l,l}=\beta_{l}$. The
large-scale fading effects are for terminal $l$ are captured
in $\beta_{l}=\varrho\zeta_{l}\left(r_{0}/r_{l}\right)^{\alpha}$.
In particular, $\varrho$ denotes the
unit-less constant for geometric attenuation at a reference distance $r_{0}$,
$r_{l}$ denotes the link distance between the BS array and terminal $l$,
$\alpha$ denotes the attenuation exponent and $\zeta_{l}$ models the effects of
shadow-fading following a log-normal density, i.e.,
$10\log_{10}\left(\zeta_{l}\right)\sim
{}\mathcal{N}\left(0,\sigma_{\textrm{sh}}^{2}\right)$,
with $\sigma_{\textrm{sh}}$ denoting the shadow-fading standard deviation.
Numerical values for the above are tabulated in
Section~\ref{NumericalResults}.
The $L\times{}1$ vector of uplink data symbols from the $L$ terminals is
given by $\bm{s}$, such that the $l$-th entry of $\bm{s}$,
$s_{l}$ has $\mathbb{E}\left[|s_{l}|^{2}\right]=1$. Additive white
Gaussian noise entries at the $M$ BS antennas is given by the $M\times{}1$
vector $\bm{n}$, such that the $l$-th
entry of $\bm{n}$, $n_{l}\sim{}\mathcal{CN}\left(0,\sigma^{2}\right)$.
We assume that $\sigma^{2}=1$, hence the average uplink
SNR, defined as $\rho/\sigma^{2}=\rho$.

\vspace{-5pt}
\subsection{Channel Model}
\label{ChannelModel}
\vspace{-3pt}
Previous studies (e.g., \cite{NGO,ZHANG,GE}) on large SC MU-MIMO systems
consider a physical channel model based on full NLoS propagation conditions,
where the BS sees the same set of
scattered directions from each terminal. We extend this model to cater for
the presence of LoS in the propagation channel, as well as a unique set of scattered
directions from each terminal taking into account differences in the local scattering
around each terminal.
Specifically, $\bm{G}=\left[\bm{g}_{1},\dots{},\bm{g}_{L}\right]$,
where $\bm{g}_{l}$, the $l$-th column of $\bm{G}$ contains the $M\times{}1$ uplink
channel vector from terminal $l$ to the BS array given by
\vspace{-5pt}
\begin{equation}
 \label{terminalgchannel}
 \bm{g}_{l}=\eta'_{l}\hspace{1pt}
 \bm{A}_{l}\hspace{1pt}\bm{h}_{l}\hspace{-1pt}+\hspace{-1pt}
 \bar{\eta}_{l}\hspace{1pt}\bar{\bm{h}}_{l},
 \vspace{-5pt}
\end{equation}
where $\eta'_{l}=\eta_{l}\frac{1}{\sqrt{P}}$ with
$\eta_{l}\hspace{-2pt}=\hspace{-2pt}\big(\frac{1}{1+K_{l}}\big)^{\hspace{-2pt}
\frac{1}{2}}$ and $\bar{\eta_{l}}\hspace{-3pt}=
\hspace{-3pt}\big(\frac{K_{l}}{K_{l}+1}\big)^{\hspace{-2pt}\frac{1}{2}}$.
In the above,
$\eta_{l}$ and $\bar{\eta}_{l}$ balance the amount of power present in the diffuse and
specular components of the channel according to the Ricean $K$-factor,
$K_{l}$, specific to terminal $l$ \cite{MOLISCH}. Moreover,
$\eta_{l}$ is further scaled by a factor of $\frac{1}{\sqrt{P}}$ to
normalize the steering vectors in $\bm{A}_{l}$, the
$M\times{}P$ receive steering matrix associated with
the diffuse components of the channel. Here, $P$ denotes a large yet finite
number of diffuse wavefronts. For ULAs
\vspace{-4pt}
\begin{equation}
 \label{correlationmatrixterminall}
 \bm{A}_{l}=\left[\bm{a}\left(\phi_{l,1}\right),
 \bm{a}\left(\phi_{l,2}\right),\dots{},\bm{a}\left(\phi_{l,P}\right)\right],
 \vspace{-3pt}
\end{equation}
where each vector in \eqref{correlationmatrixterminall} is given by
\vspace{-2pt}
\begin{equation}
 \label{aphili}
 \bm{a}\left(\phi_{l,i}\right)=\left[1,e^{j2\pi{}d\sin\left(\phi_{l,i}\right)}
 ,\dots{},e^{j2\pi{}\left(M-1\right)d\sin\left(\phi_{l,i}\right)}\right].
\vspace{-2pt}
 \end{equation}
We note that $i\in\left\{1,\dots,P\right\}$, with $d$ denoting the equidistant
inter-element spacing normalized by the carrier wavelength, $\lambda$;
$\phi_{l,i}\in\left[-\Delta/2,\Delta/2\right]$ denotes the
$i$-th direction-of-arrival (DOA)
from terminal $l$ to the BS array and $\Delta$ is the angular spread in
the azimuth domain. With such a model, the angular spread can be
modeled by having a large $P$, whilst different degrees of receive correlation
are adjusted by varying the angular spread. Moreover, $\bm{h}_{l}\sim{}\mathcal{CN}\left(
0,\bm{I}_{P}\right)$ is the $P\times{}1$ vector of diffuse channel gains, whilst
$\bar{\bm{h}}_{l}$ is the $M\times{}1$ vector denoting the specular component of the
channel and is governed by the ULA's steering response with a LoS DoA,
$\bar{\phi}_{l}$ for terminal $l$, such that
\vspace{-3pt}
\begin{equation}
 \label{hlosterminall}
 \bar{\bm{h}}_{l}=\left[1,e^{j2\pi{}d\sin\left(\bar{\phi}_{l}\right)},\dots{},
 e^{j2\pi{}\left(M-1\right)d\sin\left(\bar{\phi_{l}}\right)}\right].
 \vspace{-3pt}
\end{equation}

\textbf{Remark 1.}
For both $\bm{a}\left(\phi_{l,i}\right)$ and $\bar{\bm{h}}_{l}$,
we note that the normalized total array length, $d_{0}$, is fixed at the BS,
such that the inter-element spacing between two successive elements is
given by $d=\frac{d_{0}}{M-1}\lambda$. Since the physical dimensions of the
BS array are predetermined, the above model accurately allows us to capture the
correlation due to close proximity of adjacent antenna elements positioned at the
array. This along with the unique correlation matrices for each terminal created by the
$\bm{A}_{l}$ for $l\in{}\left\{1,\dots{},L\right\}$ constitutes our focus in
the following sections. We note that in this study, we neglect the effects of antenna
coupling, since they can be compensated by impedance matching techniques
as shown in \cite{WARNICK,GARCIA-ROD}.

To determine the level of LoS and NLoS present in the propagation channel from a
given terminal to the BS, we employ a probability based approach following
\cite{TATARIA}. Both LoS and NLoS probabilities are a function of
the link distance, from which the LoS and NLoS geometric attenuation, as well as
other link characteristics are obtained. We consider propagation parameters
from both microwave \cite{3GPPTR36873} and mmWave \cite{AKDNENIZ} frequency bands.
For notational clarity, we delay the discussion of the above mentioned parameters to
Section~\ref{NumericalResults}.

\vspace{-3pt}
\subsection{Per-Terminal SINR and Ergodic Sum Spectral Efficiency}
\vspace{-2pt}
As linear signal processing techniques perform near optimally for large
MU-MIMO systems \cite{RUSEK,LARSSON}, we employ a linear receiver in the form
of a MRC at the BS. The $L\times{}M$ MRC matrix, $\bm{G}^{\textrm{H}}$, is used to
separate $\bm{y}$ into $L$ streams by
\begin{equation}
 \label{receivedsignalafterMRC}
 \bm{r}=\bm{G}^{\textrm{H}}\bm{y}=\rho^{\frac{1}{2}}\bm{G}^{\textrm{H}}\bm{G}
 \bm{D}^{\frac{1}{2}}\bm{s}+\bm{G}^{\textrm{H}}\bm{n}.
\end{equation}
Thus, the detected signal from terminal $l$ is given by
\begin{equation}
 \label{receivedsignalterminall}
 r_{l}=\rho^{\frac{1}{2}}\beta_{l}^{\frac{1}{2}}
 \bm{g}_{l}^{\textrm{H}}\bm{g}_{l}s_{l}\hspace{-1pt}+\hspace{-1pt}
 \rho^{\frac{1}{2}}\sum\nolimits_{\substack{k=1\\k\neq{}l}}^{L}
 \hspace{-1pt}\beta_{k}^{\frac{1}{2}}\bm{g}_{l}^{\textrm{H}}
 \bm{g}_{k}s_{k}\hspace{-1pt}+\hspace{-1pt}
 \bm{g}_{l}^{\textrm{H}}\bm{n},
 \end{equation}
 resulting in the corresponding SINR given by
\begin{equation}
 \label{SINRterminall}
 \textrm{SINR}_{l}=\frac{\rho\beta_{l}||\bm{g}_{l}||^{4}}{||\bm{g}_{l}||^{2}+
 \rho\sum\nolimits_{\substack{k=1\\k\neq{}l}}^{L}\beta_{k}|\bm{g}_{l}^{\textrm{H}}
 \bm{g}_{k}|^{2}}.
\end{equation}
Hence, the instantaneous achievable uplink spectral efficiency for
terminal $l$ (measured in bits/sec/Hz) can be computed as $\textrm{R}_{l}=
\log_{2}\left(1+\textrm{SINR}_{l}\right)$. As such,
the ergodic sum spectral efficiency over all $L$
terminals is given by
\begin{equation}
 \label{ergodicsumse}
 \mathbb{E}\left[\textrm{R}_{\textrm{sum}}\right]=\mathbb{E}\left[
 \sum\nolimits_{l=1}^{L}\hspace{-1pt}\textrm{R}_{l}\right],
\end{equation}
where the expectation is performed over the fast-fading. In
the following section, we derive tight analytical expressions to approximate
the expected value of \eqref{SINRterminall} and \eqref{ergodicsumse}, respectively.

\vspace{3pt}
\section{Expected Per-Terminal SINR and Ergodic Sum Spectral Efficiency Analysis}
\label{ExpectedPerTerminalSINRandErgodicSumSpectralEff}
\vspace{3pt}
The expected SINR for terminal $l$ can be obtained by taking the expectation of
the ratio in \eqref{SINRterminall}. However, exact evaluation of this is
extremely cumbersome \cite{ZHANG2,BASNAYAKA}. Hence, we resort to the commonly
used first-order Delta expansion, as shown in \cite{ZHANG2,BASNAYAKA} and
references therein. This gives
\begin{equation}
 \label{expectedSINRterminall}
 \mathbb{E}\left[\textrm{SINR}_{l}\right]\approx
 \frac{\rho\beta_{l}\mathbb{E}\left[||\bm{g}_{l}||^{4}\right]}
 {\mathbb{E}\left[||\bm{g}_{l}||^{2}\right]+
 \rho\sum\nolimits_{\substack{k=1\\k\neq{}l}}^{L}\beta_{k}
 \mathbb{E}\left[|\bm{g}_{l}^{\textrm{H}}\bm{g}_{k}|^{2}\right]}.
\end{equation}

\textbf{Remark 2.} The approximation in \eqref{expectedSINRterminall} is of the
form of $\mathbb{E}\left[\frac{X}{Y}\right]\approx\frac{\mathbb{E}\left[X\right]}
{\mathbb{E}\left[Y\right]}$. The accuracy of such approximations relies on $Y$
having a small variance relative to its mean. This can be seen by applying a
multivariate Taylor series expansion of $\frac{X}{Y}$ around
$\frac{\mathbb{E}\left[X\right]}{\mathbb{E}\left[Y\right]}$, as shown in the
analysis methodology of \cite{ZHANG2}. In particular, both $X$ and $Y$ are
well suited to this approximation as $M$ and $L$ start to increase (the case for
large MU-MIMO systems), where the approximation is shown to be extremely
tight. This is due to $X$ and $Y$ averaging their respective individual
components, minimizing their variance relative to their mean. For further
discussion, we refer the interested reader to Appendix I of \cite{ZHANG2},
where a detailed mathematical proof of the approximation accuracy can be found.

In the sequel, Lemmas 1, 2 and 3 derive the expectations in the numerator and
denominator of \eqref{expectedSINRterminall}.

\textbf{Lemma 1.} For a ULA with $M$ antennas in a fixed physical
space at the BS, considering a correlated Ricean fading channel in
$\bm{g}_{l}$ from terminal $l$ to the BS
\vspace{-4pt}
\begin{equation}
\nonumber
\hspace{-38pt}
\delta_{l}=
\mathbb{E}\left[||\bm{g}_{l}||^{4}\right]={\left(\eta'_{l}\right)}^{4}\left\{P^{2}
M^{2}+\textrm{tr}\left[\left(\bm{A}_{l}^{\textrm{H}}
\bm{A}_{l}\right)^{2}\right]\right\}+
\vspace{-4pt}
\end{equation}
\begin{equation}
\label{lemma1}
\hspace{-1pt}
2PM^{2}\hspace{-2pt}\left(\eta'_{l}\right)^{2}\hspace{-2pt}
\left(\bar{\eta}_{l}\right)^{2}
+\hspace{2pt}2\left(\eta'_{l}\right)^{2}\hspace{-2pt}\left(\bar{\eta}_{l}\right)^{2}
\bm{\bar{h}}_{l}^{\textrm{H}}\bm{A}_{l}\bm{A}_{l}^{\textrm{H}}\bm{\bar{h}}_{l}
+\left(\bar{\eta}_{l}\right)^{4}\hspace{-2pt}M^{2},
\vspace{-4pt}
\end{equation}
where each parameter is defined after \eqref{terminalgchannel}.

\emph{Proof:} See Appendix~\ref{ProofLemma1}. \hfill\ensuremath{\blacksquare}

\textbf{Lemma 2.} Under the same conditions as Lemma 1,
\vspace{-4pt}
\begin{equation}
\nonumber
\varphi_{l,k}=\mathbb{E}\left[|\bm{g}_{l}^{\textrm{H}}\bm{g}_{k}|^{2}\right]=
\left(\eta'_{l}\right)^{2}\hspace{-2pt}
\left(\eta'_{k}\right)^{2}\textrm{tr}\left[\hspace{-1pt}\bm{A}_{k}
\hspace{-1pt}\bm{A}_{k}^{\textrm{H}}\hspace{-1pt}
\bm{A}_{l}\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}\right]+
\left(\eta'_{l}\right)^{2}\hspace{-2pt}\left(\bar{\eta}_{k}\right)^{2}
\vspace{-2pt}
\end{equation}
\begin{equation}
\nonumber
\hspace{-16pt}
\textrm{tr}\left[\bar{\bm{h}}_{k}^{\textrm{H}}\hspace{-1pt}
\bm{A}_{l}\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}\hspace{-1pt}
\bar{\bm{h}}_{k}\right]\hspace{-2pt}+\hspace{-1pt}
\left(\bar{\eta}_{l}\right)^{2}\left(\eta'_{k}\right)^{2}
\hspace{-2pt}\textrm{tr}\left[\bar{\bm{h}}_{l}^{\textrm{H}}
\bm{A}_{k}\hspace{-1pt}\bm{A}_{k}^{\textrm{H}}
\hspace{-1pt}\bar{\bm{h}}_{l}\right]\hspace{-1pt}+\hspace{-1pt}
\left(\bar{\eta}_{l}\right)^{\hspace{-1pt}2}
\left(\bar{\eta}_{k}\right)^{\hspace{-1pt}2}
\end{equation}
\begin{equation}
\hspace{-200pt}
 \label{lemma2}
 \bar{\bm{h}}_{l}^{\textrm{H}}
 \bar{\bm{h}}_{k}\bar{\bm{h}}_{k}^{\textrm{H}}
 \bar{\bm{h}}_{l}.
 \vspace{-4pt}
\end{equation}

\emph{Proof:} See Appendix~\ref{ProofLemma2}. \hfill\ensuremath{\blacksquare}

\textbf{Lemma 3.} Under the same conditions as Lemma 1,
\vspace{-4pt}
\begin{equation}
\label{lemma3}
\chi_{l}=\mathbb{E}\left[||\bm{g}_{l}||^{2}\right]=M\left[P\left(\eta'_{l}\right)^{2}
+\left(\bar{\eta}_{l}\right)^{2}\right].
\vspace{-4pt}
\end{equation}

\emph{Proof:} We begin by substituting the definition of $\bm{g}_{l}$
into $\chi_{l}$ and expanding, allowing us to write
\vspace{-2pt}
\begin{equation}
\label{lemma3proof31}
\chi_{l}\hspace{-2pt}=\hspace{-2pt}\mathbb{E}\left[||\bm{g}_{l}||^{2}\right]
\hspace{-3pt}=\hspace{-2pt}
\mathbb{E}\left[\left(\eta'_{l}\right)^{2}\hspace{-2pt}\bm{h}_{l}^{\textrm{H}}
\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}\bm{h}_{l}\right]\hspace{-1pt}+
\hspace{-1pt}\mathbb{E}\left[\left(\bar{\eta}_{l}\right)^{2}\hspace{-1pt}
\bar{\bm{h}}_{l}^{\textrm{H}}\bar{\bm{h}}_{l}\right].
\vspace{-3pt}
\end{equation}
Performing the expectations with respect to $\bm{h}_{l}$ and
extracting the relevant constants yields
\vspace{-2pt}
\begin{equation}
\label{lemma3proof32}
\chi_{l}=\mathbb{E}\left[||\bm{g}_{l}||^{2}\right]\hspace{-2pt}=
\hspace{-2pt}\left(\eta'_{l}\right)^{2}\textrm{tr}\left[
\bm{A}_{l}^{\textrm{H}}
\bm{A}_{l}\right]\hspace{-2pt}+\hspace{-2pt}
\left(\bar{\eta}_{l}\right)^{2}
\mathbb{E}\left[\bar{\bm{h}}_{l}^{\textrm{H}}\bar{\bm{h}}_{l}\right].
\vspace{-1pt}
\end{equation}
Recognizing that $\textrm{tr}\left[\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}
\right]=PM$ and $\mathbb{E}\left[\bar{\bm{h}}_{l}^{\textrm{H}}
\bar{\bm{h}}_{l}\right]=M$ allows us to state
\vspace{-5pt}
\begin{equation}
\label{lemma3proof33}
\chi_{l}=\mathbb{E}\left[||\bm{g}_{l}||^{2}\right]=
M\left[P\left(\eta'_{l}\right)^{2}+\left(\bar{\eta}_{l}\right)^{2}\right],
\vspace{-5pt}
\end{equation}
concluding the proof. \hfill\ensuremath{\blacksquare}

\textbf{Theorem 1.} With MRC at the BS consisting of a
space-constrained ULA, the expected uplink SINR of terminal $l$
in a spatially correlated Ricean fading channel can be
approximated as
\vspace{-3pt}
\begin{equation}
\label{theorem1}
\mathbb{E}\left[\textrm{SINR}_{l}\right]\approx
\frac{\rho\beta_{l}\delta_{l}}{\chi_{l}+\rho{}
\sum\nolimits_{\substack{k=1\\k\neq{}l}}^{L}
\beta_{k}\hspace{1pt}\varphi_{l,k}}.
\vspace{-2pt}
\end{equation}

\emph{Proof:} Substituting the results from Lemmas 1, 2 and 3
for $\delta_{l},\varphi_{l,k}$ and $\chi_{l}$ yields the desired
expression in \eqref{theorem1}. \hfill\ensuremath{\blacksquare}

\textbf{Remark 3.} The result in \eqref{theorem1} is extremely general and
is a closed-form solution to a complex scenario, where in addition to fixed
physical spacing and MRC at the BS, each terminal has a unique LoS direction,
unique Rice factor, unique receive correlation matrix and a
unique link gain. It can be readily observed via inspection,
that both the numerator and the
denominator of \eqref{theorem1} are influenced by each of the above factors.
The result allows for a general evaluation of large MU-MIMO systems
with space-constrained ULAs and lends itself to many useful
special cases (as shown in Section~\ref{SpecialCases}).

We note that \eqref{theorem1} can be further used to approximate
the ergodic sum spectral efficiency of the system by
\begin{equation}
 \label{approxergse}
 \mathbb{E}\left[\textrm{R}_\textrm{sum}\right]\approx
\sum\nolimits_{l=1}^{L}\log_{2}\left(1+\mathbb{E}\left[
\textrm{SINR}_{l}\right]\right).
\end{equation}
The accuracy of the derived closed-form expressions in \eqref{theorem1} and
\eqref{approxergse} is demonstrated in Section~\ref{NumericalResults}.
In the following section, we present three special cases of Theorem 1
demonstrating its generality.

\vspace{-5pt}
\section{Special Cases}
\label{SpecialCases}
\textbf{Corollary 1.} With MRC at the BS consisting of a SC ULA,
the expected uplink SINR of terminal $l$ with
no LoS, i.e, Rayleigh fading with unequal correlation matrices for
each terminal, can be approximated as
\vspace{4pt}
\begin{equation}
 \nonumber
 \hspace{-190pt}
\mathbb{E}\left[\textrm{SINR}_{l}^{\textrm{C1}}\right]
\hspace{-3pt}\approx\hspace{-2pt}
\vspace{-7pt}
\end{equation}
\begin{equation}
 \label{corollary1}
 \hspace{8pt}
 \frac{\rho\beta_{l}\left(\eta'_{l}\right)^{4}\left\{
 P^{2}M^{2}\hspace{-1pt}+\hspace{-1pt}
 \textrm{tr}\left[\left(\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}
 \right)^{2}\right]\right\}}
 {M\hspace{-2pt}P\hspace{-2pt}\left(\eta'_{l}\right)^{2}\hspace{-2pt}+
 \hspace{-2pt}\rho
 \sum\nolimits_{\substack{k=1\\k\neq{}l}}^{L}
 \left\{\beta_{k}\left(\eta'_{l}
 \right)^{2}\hspace{-1pt}\left(\eta'_{k}\right)^{2}
 \hspace{-1pt}\textrm{tr}\left[
 \hspace{-1pt}\bm{A}_{k}\hspace{-1pt}\bm{A}_{k}^{\textrm{H}}
 \hspace{-1pt}\bm{A}_{l}\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}
 \right]\right\}}.
 \vspace{-2pt}
\end{equation}

\emph{Proof:} Substituting $\delta_{l}$, $\chi_{l}$ and
$\varphi_{l}$ into \eqref{theorem1} and setting
$\bar{\eta}_{l}\hspace{-1pt}=\hspace{-1pt}
\bar{\eta}_{k}=0$, $\eta'_{l}\hspace{-1pt}=\hspace{-1pt}
\eta'_{k}=\frac{1}{\sqrt{P}}$ as $K_{l}=0$ and
$\bar{\bm{h}}_{l}=\bar{\bm{h}}_{k}=\bm{0}_{M\times{}1}$,
where $\bm{0}_{M\times{}1}$ denotes a $M\times{}1$ vector of
zeros for $l,k\in\left\{1,\dots{},L\right\}$ yielding
the desired expression. \hfill\ensuremath{\blacksquare}

\vspace{2pt}
\textbf{Corollary 2 (Proposition 1 in \cite{ZHANG}).}
With MRC processing at the BS containing of a SC
ULA, the expected uplink SINR for terminal $l$ with
no LoS and equal correlation matrices, i.e.,
Rayleigh fading with a fixed correlation for each
terminal, can be approximated as
\vspace{4pt}
\begin{equation}
\nonumber
\hspace{-190pt}
 \mathbb{E}\left[\textrm{SINR}_{l}^{\textrm{C2}}\right]
\hspace{-3pt}\approx\hspace{-2pt}
\vspace{-7pt}
\end{equation}
\begin{equation}
 \label{corollary2}
 \frac{\rho\beta_{l}\left(\eta'_{l}\right)^{4}\left\{
 P^{2}M^{2}\hspace{-1pt}+\hspace{-1pt}
 \textrm{tr}\left[\left(\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}
 \right)^{2}\right]\right\}}
 {M\hspace{-2pt}P\hspace{-2pt}\left(\eta'_{l}\right)^{2}\hspace{-2pt}+
 \hspace{-2pt}\rho
 \sum\nolimits_{\substack{k=1\\k\neq{}l}}^{L}
 \left\{\beta_{k}\left(\eta'_{l}
 \right)^{2}\hspace{-1pt}\left(\eta'_{k}\right)^{2}
 \textrm{tr}\left[\left(\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}
 \right)^{2}\right]\right\}}.
 \vspace{-3pt}
\end{equation}

\emph{Proof:} Following the approach outlined in the proof of
Corollary 1 and recognizing that $\textrm{tr}\big[\hspace{-2pt}\left(
\hspace{-1pt}\bm{A}_{l}\hspace{-1pt}
\bm{A}_{l}^{\textrm{H}}\right)^{2}\big]=\textrm{tr}
\left[\bm{A}_{l}\hspace{-1pt}
\bm{A}_{l}^{\textrm{H}}\hspace{-1pt}
\bm{A}_{l}\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}
\right]\hspace{-2pt}=\hspace{-2pt}\textrm{tr}
\big[\hspace{-1pt}\left(\hspace{-1pt}
\bm{A}_{l}^{\textrm{H}}\hspace{-1pt}\bm{A}_{l}\right)^{2}\big]$
yields the desired result. \hfill\ensuremath{\blacksquare}

\vspace{2pt}
\textbf{Corollary 3.} With MRC at the BS consisting of a SC ULA,
the expected uplink SINR of terminal $l$ with
LoS i.e., correlated Ricean fading, with equal correlation
matrices for each terminal can be approximated as
\begin{equation}
\label{corollary3}
\mathbb{E}\left[\textrm{SINR}_{l}^{\textrm{C3}}\right]\approx
\frac{\rho\beta_{l}\delta_{l}}{\chi_{l}+\rho
\sum\nolimits_{\substack{k=1\\k\neq{}l}}^{L}
\beta_{k}\varphi'_{l,k}},
\vspace{-10pt}
\end{equation}
where
\vspace{-2pt}
\begin{equation}
\nonumber
\varphi'_{l,k}=\left(\eta'_{l}\right)^{2}\hspace{-2pt}
\left(\eta'_{k}\right)^{2}\textrm{tr}\big[\hspace{-1pt}
\left(\hspace{-2pt}\bm{A}_{l}
\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}\right)^{2}\big]\hspace{-2pt}+\hspace{-2pt}
\left(\eta'_{l}\right)^{2}\hspace{-2pt}\left(\bar{\eta}_{k}\right)^{2}
\textrm{tr}\left[\bar{\bm{h}}_{k}^{\textrm{H}}\hspace{-1pt}
\bm{A}_{l}\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}\hspace{-1pt}
\bar{\bm{h}}_{k}\right]
\vspace{-3pt}
\end{equation}
\begin{equation}
\label{varphiprimel}
\hspace{-52pt}
\hspace{-2pt}+\hspace{-1pt}
\left(\bar{\eta}_{l}\right)^{2}\left(\eta'_{k}\right)^{2}
\hspace{-1pt}\textrm{tr}\left[\bar{\bm{h}}_{l}^{\textrm{H}}\hspace{-2pt}
\bm{A}_{l}\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}\bar{\bm{h}}_{l}\right]
\hspace{-2pt}+\hspace{-2pt}\left(\bar{\eta}_{l}\right)^{\hspace{-1pt}2}
\left(\bar{\eta}_{k}\right)^{\hspace{-1pt}2}M^{2}.
\vspace{-5pt}
\end{equation}

\emph{Proof:} Replacing $\textrm{tr}\left[\bm{A}_{k}\hspace{-1pt}
\bm{A}_{k}^{\textrm{H}}\hspace{-1pt}\bm{A}_{l}\hspace{-1pt}
\bm{A}_{l}^{\textrm{H}}\right]$ with
$\textrm{tr}\big[\hspace{-2pt}
\left(\hspace{-2pt}\bm{A}_{l}\bm{A}_{l}^{\textrm{H}}\right)^{2}
\big]$ yields the desired expression in \eqref{corollary3}.
We note that $\delta_{l}$ and $\chi_{l}$ are as
defined in \eqref{lemma1} and \eqref{lemma3}, respectively. \hfill\ensuremath{\blacksquare}

\textbf{Remark 4.} Corollaries 1 and 2 share
a common trend in that both the numerators and denominators are
governed by spatial correlation matrices in $\bm{A}_{l}$ and $\bm{A}_{k}$,
respectively. In the case where correlation matrices are fixed for each
terminal, the trace in their respective denominators
can be readily seen to translate from
$\textrm{tr}\left[\bm{A}_{k}\hspace{-1pt}\bm{A}_{k}^{\textrm{H}}
\hspace{-1pt}\bm{A}_{l}\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}\right]$ to
$\textrm{tr}\left[(\bm{A}_{l}^{\textrm{H}}\hspace{-1pt}\bm{A}_{l})^{2}\right]$.

In the subsequent section, we analyze the convergence of the expected
per-terminal SINR and ergodic spectral efficiency with MRC,
as the number of receive antennas, $M$, grows without bound with a fixed
number of user terminals, $L$.

\section{Limiting Expected Per-Terminal SINR and Ergodic Sum
Spectral Efficiency Analysis}
\label{LimitingPerTerminalSINRandErgodicSumSpectralEff}
Theorem 1 presents an expected uplink SINR approximation for terminal $l$
which is suitable for any system size, as well as
any operating SNR, LoS level, spatial correlation level and
physical array spacing. We now examine the
asymptotic behavior of \eqref{theorem1}, as $M\rightarrow\infty$, with a
fixed (finite) $L$. Dividing through by $M^{2}$ throughout, we observe
the limit as
\begin{equation}
 \label{limitingSINR2}
 \overline{\mathbb{E}\left[\textrm{SINR}_{l}\right]}=
 \lim_{M\rightarrow{}\infty}
  \Bigg\{\frac{\rho\beta_{l}\left(\delta_{l}/M^{2}\right)}
 {\left(\chi_{l}/M^{2}\right)+
 \rho\sum\nolimits_{\substack{k=1\\k\neq{}l}}^{L}
 \beta_{k}\left(\varphi_{l,k}/M^{2}\right)}\Bigg\}.
 \vspace{-6pt}
\end{equation}
Referring to the numerator of \eqref{limitingSINR2}, two
terms in
\vspace{-1pt}
\begin{equation}
 \delta_{l}^{1}=\left(\eta'_{l}\right)^{4}
(\textrm{tr}[(\bm{A}_{l}^{\textrm{H}}
\hspace{-1pt}\bm{A}_{l})^{2}]/M^{2}),
\vspace{-3pt}
\end{equation}
and
\begin{equation}
\vspace{-3pt}
 \delta_{l}^{2}=2\left(\eta'_{l}\right)^{2}
\left(\bar{\eta}_{l}\right)^{2}
(\bar{\bm{h}}_{l}^{\textrm{H}}
\bm{A}_{l}\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}
\bar{\bm{h}}_{l}/M^{2}),
\end{equation}
do not vanish from $\delta_{l}$ as $M$ grows without bound,
whilst the denominator of \eqref{limitingSINR2} has four
terms, these are
\vspace{-1pt}
\begin{equation}
 \varphi_{l,k}^{1}=\left(\eta'_{l}\right)^{2}
\left(\eta'_{k}\right)^{2}(\textrm{tr}\left[\bm{A}_{k}
\bm{A}_{k}^{\textrm{H}}\bm{A}_{l}
\bm{A}_{l}^{\textrm{H}}\right]\hspace{-2pt}/M^{2}),
\vspace{-3pt}
\end{equation}
\begin{equation}
 \varphi_{l,k}^{2}=
\left(\eta'_{l}\right)^{2}
\left(\bar{\eta}_{k}\right)^{2}(\textrm{tr}\left[
\bar{\bm{h}}_{k}^{\textrm{H}}
\bm{A}_{l}\bm{A}_{l}^{\textrm{H}}\bar{\bm{h}}_{k}\right]/M^{2}),
\vspace{-3pt}
\end{equation}
\begin{equation}
 \varphi_{l,k}^{3}=
\left(\bar{\eta}_{l}\right)^{2}\left(\eta'_{k}\right)^{2}
(\textrm{tr}\left[\bar{\bm{h}}_{l}^{\textrm{H}}\bm{A}_{k}
\hspace{-2pt}\bm{A}_{k}^{\textrm{H}}
\bar{\bm{h}}_{l}\right]/M^{2}),
\vspace{-3pt}
\end{equation}
and
\vspace{-3pt}
\begin{equation}
 \varphi_{l,k}^{4}=\left(\bar{\eta}_{l}\right)^{2}
\left(\bar{\eta}_{k}\right)^{2}(
\bar{\bm{h}}_{l}^{\textrm{H}}\bar{\bm{h}}_{k}
\bar{\bm{h}}_{k}^{\textrm{H}}\bar{\bm{h}}_{l}/M^{2}),
\vspace{-4pt}
\end{equation}
which do not vanish from $\varphi_{l,k}$ as $M\rightarrow\infty$.

In the sequel, Lemmas 4, 5 and 6 derive the limiting value of (24)-(29),
respectively.

\textbf{Lemma 4.} $\lim\limits_{M\rightarrow{}\infty}\varphi_{l,k}^{4}$
is given by
\vspace{-8pt}
\begin{equation}
 \nonumber
 \bar{\varphi}_{l,k}^{4}=
 \left(\bar{\eta}_{l}\right)^{2}
 \left(\bar{\eta}_{k}\right)^{2}
  \lim_{M\rightarrow{}\infty}\left\{\hspace{2pt}\left|
 \frac{\bar{\bm{h}}_{l}^{\textrm{H}}\bar{\bm{h}}_{k}}
 {M}\right|^{2}\right\}
 \vspace{-3pt}
\end{equation}
\begin{equation}
 \label{lemma4}
 \hspace{-8pt}
 =\left(\bar{\eta}_{l}\right)^{2}
 \left(\bar{\eta}_{k}\right)^{2}
 \vartheta\left(\bar{\phi}_{l},
 \bar{\phi}_{k}\right)^{2},
 \vspace{-3pt}
\end{equation}
where $\vartheta\left(\bar{\phi}_{l},\bar{\phi}_{k}
\right)=\left|\textrm{sinc}\left(\pi{}d_{0}\left(
\sin\left(\bar{\phi}_{l}\right)-
\sin\left(\bar{\phi}_{k}\right)\right)\big/\lambda\right)
\right|,$ where $\textrm{sinc}\left(\cdot\right)$ denotes the
sinc function.

\vspace{1pt}
\emph{Proof:} We begin by defining
\vspace{-2pt}
\begin{equation}
 \nonumber
 \hspace{-120pt}
 \vartheta\left(\bar{\phi}_{l},\bar{\phi}_{k}
 \right)=\lim_{M\rightarrow{}\infty}\left\{\left|
 \frac{\bar{\bm{h}}_{l}^{\textrm{H}}\bar{\bm{h}}_{k}}
 {M}\right|\right\}
 \vspace{-3pt}
\end{equation}
\begin{equation}
\nonumber
\hspace{40pt}
 =\lim_{M\rightarrow\infty}\left\{\left|
 \frac{1}{M}\sum\nolimits_{c=0}^{M-1}e^{j2\pi{}\frac{c}
 {\lambda}\frac{d_{0}}{M-1}\left(\sin\left(\bar{\phi_{l}}
 \right)-\sin\left(\bar{\phi_{k}}\right)\right)}\right|\right\}
\vspace{-3pt}
 \end{equation}
\begin{equation}
\nonumber
\hspace{-18pt}
 =\left|\int\nolimits_{0}^{1}e^{j2\pi{}\frac{d_{0}}
 {\lambda}\left(\sin\left(\bar{\phi}_{l}\right)
 -\sin\left(\bar{\phi}_{k}\right)\right)f}df\right|
 \vspace{-3pt}
 \end{equation}
\begin{equation}
 \label{lemma4proof1}
 \hspace{5pt}
 =|\textrm{sinc}\left(\pi{}d_{0}\left(\sin\left(
 \bar{\phi_{l}}\right)-\sin\left(\bar{\phi}_{k}
 \right)\right)/\lambda\right)|,
 \vspace{-5pt}
\end{equation}
yielding the desired result. \hfill\ensuremath{\blacksquare}

\textbf{Remark 5.} The expression in \eqref{lemma4} is another
closed-form solution and can be readily seen to be
dependent on the respective LoS angles unique to terminals $l$ and $k$.

\textbf{Lemma 5.} $\lim\limits_{M\rightarrow\infty}\varphi_{l,k}^{3}$ is
given by
\vspace{-5pt}
\begin{equation}
 \nonumber
 \bar{\varphi}_{l,k}^{3}=
 \left(\bar{\eta}_{l}\right)^{2}
 \left(\eta'_{k}\right)^{2}
 \lim_{M\rightarrow{}\infty}
 \left\{\frac{\textrm{tr}
 \left[\bar{\bm{h}}_{l}^{\textrm{H}}
 \bm{A}_{k}\bm{A}_{k}^{\textrm{H}}\bar{\bm{h}}_{l}\right]}{M^{2}}
 \right\}
 \vspace{-8pt}
\end{equation}
\begin{equation}
 \label{lemma5}
 \hspace{-16pt}
 =\left(\bar{\eta}_{l}\right)^{2}\hspace{-2pt}
 \left(\eta'_{k}\right)^{2}
 \sum\limits_{r=1}^{P}
 \vartheta\left(\bar{\phi}_{l},\phi_{k,r}\right)^{2}.
 \vspace{-4pt}
\end{equation}

\emph{Proof:} Using similar methodology as in the proof
of Lemma 4, we recognize that
\vspace{-5pt}
\begin{equation}
 \label{lemma5proof1}
 \frac{1}{M^{2}}\hspace{2pt}\textrm{tr}\left[\bm{h}_{l}^{\textrm{H}}\bm{A}_{k}
 \bm{A}_{k}^{\textrm{H}}\bm{h}_{l}\right]
 \hspace{-2pt}=\hspace{-2pt}
 \frac{1}{M^{2}}\sum\limits_{r=1}^{P}\left|
 \bar{\bm{h}}_{l}^{\textrm{H}}\bm{A}_{k}\right|^{2}.
 \vspace{-5pt}
\end{equation}
Substituting the specular and diffuse angles, $\bar{\phi}_{l}$ and
$\phi_{k,r}$ with $r\in\left\{1,\dots{},P\right\}$, corresponding to
$\bar{\bm{h}}_{l}$ and $\bm{A}_{k}$, yields \eqref{lemma5proof1}. \hfill\ensuremath{\blacksquare}

\textbf{Remark 6.} We note that as $\varphi_{l,k}^{2}$ and
$\delta_{l}^{2}$ have a similar
structure to $\varphi_{l,k}^{3}$,
the limiting values of $\varphi_{l,k}^{2}$ and $\delta_{l}^{2}$ in
$\bar{\varphi}_{l,k}^{2}$ and $\bar{\delta}_{l}^{2}$ have the
same form as \eqref{lemma5}, except the angles in
$\vartheta\left(\cdot\right)$ are replaced with $\bar{\phi}_{k}$,
$\phi_{l,r}$ for $\bar{\varphi}_{l,k}^{2}$ and $\bar{\phi}_{l}$, $\phi_{l,r}$
for $\bar{\delta}_{l}^{2}$, respectively. We further note that both
$\bar{\varphi}_{l,k}^{2}$ and $\bar{\delta}_{l}^{2}$ will need
to have the necessary scaling of $\left(\eta'_{l}\right)^{2}\left(\bar{\eta}_{k}
\right)^{2}$ and $2\left(\eta'_{l}\right)^{2}\left(\bar{\eta_{l}}
\right)^{2}$ as shown in (27) and (25).

\textbf{Lemma 6.} $\lim\limits_{M\rightarrow{}\infty}\varphi_{l,k}^{1}$
is given by
\vspace{-6pt}
\begin{equation}
 \nonumber
 \hspace{-17pt}
 \bar{\varphi}_{l,k}^{1}=\left(\eta'_{l}\right)^{2}
 \left(\eta'_{k}\right)^{2}\lim_{M\rightarrow{}\infty}
 \left\{\frac{\textrm{tr}\left[\bm{A}_{k}\bm{A}_{k}^{\textrm{H}}
 \bm{A}_{l}\bm{A}_{l}^{\textrm{H}}\right]}{M^{2}}\right\}
 \vspace{-6pt}
\end{equation}
\begin{equation}
 \label{lemma6}
 \hspace{-20pt}
  =\left(\eta'_{l}\right)^{2}
 \left(\eta'_{k}\right)^{2}\sum\limits_{r=1}^{P}
 \sum\limits_{t=1}^{P}\vartheta\left(\phi_{k,r},
 \phi_{l,t}\right)^{2}.
\end{equation}

\vspace{-2pt}
\emph{Proof:} Manipulating the trace in \eqref{lemma6}
allows us to state
\vspace{-2pt}
\begin{equation}
 \nonumber
 \frac{1}{M^{2}}\left\{\textrm{tr}\left[\bm{A}_{k}\bm{A}_{k}^{\textrm{H}}
 \bm{A}_{l}\bm{A}_{l}^{\textrm{H}}\right]\right\}=
 \frac{1}{M^{2}}\left\{\textrm{tr}\left[\bm{A}_{k}^{\textrm{H}}
 \bm{A}_{l}\bm{A}_{l}^{\textrm{H}}\bm{A}_{k}\right]\right\}
 \vspace{-3pt}
\end{equation}
\begin{equation}
 \label{prooflemma61}
 \hspace{-60pt}
 \hspace{-1pt}=\hspace{-1pt}\frac{1}{M^{2}}
 \sum\limits_{r=1}^{P}\sum\limits_{t=1}^{P}
 \left|\bm{a}^{\textrm{H}}\left(\phi_{k,r}\right)
 \bm{a}\left(\phi_{l,t}\right)\right|^{2}\hspace{-3pt}.
 \vspace{-3pt}
\end{equation}
Substituting the respective angles and performing some routine
algebra yields the desired result. \hfill\ensuremath{\blacksquare}

\textbf{Remark 7.} We note that as $\delta_{l}^{1}$ has a similar
form to $\varphi_{l,k}^{1}$. Using the same methodology as in
Lemma 6, we can obtain $\bar{\delta}_{l}^{1}$, the limiting value
of $\delta_{1}^{1}$, where the angles in
$\vartheta\left(\cdot\right)$ are replaced by $\phi_{l,r},\phi_{l,t}$
with $\left(\eta'_{l}\right)^{4}$ providing the required scaling.

\textbf{Theorem 2.} The limiting uplink SINR for terminal $l$
with MRC and a SC ULA at the BS can be written as
\begin{equation}
 \label{theorem2}
 \overline{\mathbb{E}\left[\textrm{SINR}_{l}\right]}=
 \frac{\rho\beta_{l}\left(\bar{\delta}_{l}^{1}\hspace{1pt}+
 \bar{\delta}_{l}^{2}\right)}
 {\rho\sum\limits_{\substack{k=1\\k\neq{}l}}^{L}\beta_{k}
 (\bar{\varphi}_{l,k}^{1}+\bar{\varphi}_{l,k}^{2}+\bar{\varphi}_{l,k}^{3}
 +\bar{\varphi}_{l,k}^{4})}.
\vspace{-6pt}
\end{equation}

\emph{Proof:} Using the results from Lemmas 4, 5, 6 and keeping in mind
Remarks 6 and 7 yields the desired expression. \hfill\ensuremath{\blacksquare}

As such the limiting ergodic sum spectral efficiency is given by
\vspace{-7pt}
\begin{equation}
 \label{limitingergse}
 \overline{\mathbb{E}\left[\textrm{R}_{\textrm{sum}}\right]}=
 \sum\limits_{l=1}^{L}\log_{2}\left(1+
 \overline{\mathbb{E}\left[\textrm{SINR}_{l}\right]}\right).
 \vspace{-5pt}
\end{equation}

In the following section, we demonstrate the accuracy of the
analysis presented in Sections~\ref{ExpectedPerTerminalSINRandErgodicSumSpectralEff},
\ref{SpecialCases} and
\ref{LimitingPerTerminalSINRandErgodicSumSpectralEff}, respectively.

\vspace{-4pt}
\section{Numerical Results}
\label{NumericalResults}
Unless otherwise specified, the parameters used in the numerical
results are specified in Table~\ref{Table1} for an UMi scenario.
The parameters for microwave and mmWave frequencies were obtained
from from \cite{3GPPTR36873} and \cite{AKDNENIZ}, respectively.
A circular cell of radius $100$ m is considered with an exclusion
radius of $r_{0}=10$ m. We assume a uniform distribution of terminals in
the cell area and consider $10^{4}$ Monte-Carlo realizations for
each result. The parameter $\varrho$ is chosen such that the fifth
percentile value of the instantaneous per-terminal SINR is 0 dB
at SNR ($\rho$) = 0 dB for the system dimensions of
$M=256$ and $L=32$.
\vspace{-3pt}
\begin{table}[!h]
\begin{center}
\scalebox{0.8}{\begin{tabular}{ccc}
\toprule
\toprule
{\textbf{Parameter}} & \multicolumn{2}{c}{\textbf{Value}} \\
\cmidrule{2-3}
{} & {\textbf{Microwave}} & {\textbf{mmWave}} \\
\midrule
Carrier frequency [GHz] & $2$ & $28$\\
LoS attenuation exponent $\left[\alpha\right]$ & 2.2  & 2  \\
NLoS attenuation exponent &3.67 &2.92 \\
LoS shadow fading standard deviation $\left[\sigma_\textrm{sh}\right]$
&3 &5.8 \\
NLoS shadow fading standard deviation & 4 & 8.7 \\
$K$-Factor mean [dB] & 9 & 12 \cite{THOMAS}\\
$K$-Factor standard deviation [dB] & 5 & 3 \cite{THOMAS} \\
\bottomrule
\end{tabular}}
\end{center}
\caption{System Parameters}
\label{Table1}
\vspace{-21pt}
\end{table}

Based on the link distance, $r_{l}$, we employ a probability based approach
in determining whether the terminal experiences LoS or NLoS conditions on
the uplink to the BS. For the microwave case, the probability of terminal $l$
experiencing LoS is governed by $P_{\textrm{LoS}}\left(r_{l}\right)=
(\min\left(18/r_{l},1\right)\big(1-e^{-r_{l}/36}
\big))+e^{-r_{l}/36}$. Naturally, the probability of the terminal
experiencing NLoS is then determined
by $P_{\textrm{NLoS}}=1-P_{\textrm{LoS}}$.
Equivalently, for the mmWave case \cite{AKDNENIZ},
$P_{\textrm{LoS}}\left(r_{l}\right)=\left(1-P_{\textrm{out}}
 \left(r_{l}\right)\right)e^{-\omega{}_{\textrm{LoS}}\hspace{2pt}r_{l}}$,
where $1/\omega_{\textrm{LoS}}=67.1$ meters and $P_{\textrm{out}}$ is the
outage probability, occurring when the attenuation in either the LoS
or NLoS states is sufficiently large. For simplicity, we set
$P_{\textrm{out}}=0$ when determining the LoS and NLoS probabilities.
Upon determining the link state of each terminal, we select the corresponding
link parameters to model the large-scale propagation effects of geometric
attenuation and shadow-fading, as specified in Table~
\ref{Table1}. We assign a
unique $K$-factor, $K_{l}$, for the $l$-th user terminal from a log-normal
distribution with the mean and standard deviation specified in
Table~\ref{Table1}. We refer to this as $K_{l}
\sim\textrm{ln}\left(\textrm{mean},\textrm{standard deviation}\right)$.

First, the accuracy of the proposed expected per-terminal SINR in
\eqref{theorem1} is examined. Fig.~\ref{expsinrvssnrF1} illustrates
the expected SINR of a given terminal as a function of $\rho$ (SNR)
for a system with $M=256$ and $L=32$, $P=50$ and $d_{0}=8\lambda$.
In addition to the microwave and
mmWave cases, we consider the correlated Rayleigh fading case
for comparison purposes. We also consider the case where each terminal
is assigned a fixed $K$-factor of $5$ dB. Three trends can be observed:
Firstly, transitioning from large to small angular spread
($\Delta\sim{}u[\frac{-\pi}{2},\frac{\pi}{2}]$
to $\Delta\sim{}u[\frac{-\pi}{16},\frac{\pi}{16}])$
tends to significantly reduce the expected SINR for all cases. This is
despite the fact that the ULA contains
very large numbers of antenna elements at the BS,
and is due to the reduction in the spatial diversity (rank) of the channel,
allowing the BS array to only see a very narrow spread of
incoming power. Secondly, increasing the mean of $K$ has an adverse effect
on the expected SINR. This is because a stronger specular component in
the channel tends to reduce the multipath diversity and in-turn reduces
its overall rank. Equivalently, this can be interpreted as an increase in
the level of inter-terminal interference leading to a lower expected
per-terminal SINR.
Third, our proposed approximations are seen to remain extremely tight
for the entire SNR range for all cases. The analytical expressions are also
seen to
remain tight for the special case presented in \eqref{corollary1}, where
each terminal undergoes Rayleigh fading with unequal correlation matrices.
Furthermore, the expected SINR in each case is seen to saturate with
growing SNR, due to the inability of the MRC to mitigate inter-terminal
interference.

\begin{figure}[!t]
 \centering
 \vspace{-15pt}
 \includegraphics[width=8.3cm]{expSINRvsSNRCR.eps}
 \vspace{-12pt}
 \caption{Expected per-terminal SINR vs. $\rho$ (SNR)
 with $M=256,L=32,P=50,d_{0}=8\lambda$.}
 \label{expsinrvssnrF1}
\end{figure}
\begin{figure}[!t]
 \centering
 \vspace{-14pt}
 \includegraphics[width=8.3cm]{ergSECDFCR.eps}
 \vspace{-8pt}
 \caption{Ergodic sum spectral efficiency CDF with $M=256, L=32$, $\rho
 \hspace{3pt}(\textrm{SNR})=10$ dB, $P=50, d_{0}=8\lambda$ and
 $\Delta\sim{}u[-\frac{\pi}{16},\frac{\pi}{16}]$ (unless specified in
 the figure).}
 \label{ergseCDFF2}
\end{figure}
\begin{figure}[!t]
 \centering
 \vspace{-10pt}
 \includegraphics[width=8.3cm]{limitsCR.eps}
 \vspace{-12pt}
 \caption{Expected Per-Terminal SINR vs. $M$ with $L=32$ at
 $\rho$ (SNR) $=10$ dB, $P=50$,
 $\Delta\sim{}u[-\frac{\pi}{16},\frac{\pi}{16}]$.}
 \label{limitsF3}
 \vspace{-20pt}
\end{figure}
Considering the special cases in \eqref{corollary2} and \eqref{corollary3},
we now examine the influence of LoS, as well as equal and unequal
correlation matrices on the ergodic sum spectral efficiency, as
shown in Fig.~\ref{ergseCDFF2}. Using the same propagation parameters from
Fig.~\ref{expsinrvssnrF1},
(listed in the figure caption) at $\rho$ (SNR) $\hspace{-3pt}=10$ dB,
we compare the cumulative distribution functions (CDFs)
of the derived ergodic sum spectral efficiency approximation in
\eqref{approxergse} with its simulated counterparts. We note that the
CDF is obtained by averaging over the
fast-fading in the channel with each value representing the variations
in the link gains and $K$-factors. We notice that
irrespective of the underlaying propagation characteristics
(Rayleigh or Ricean fading), unequal correlation matrices results in a
higher ergodic sum spectral efficiency of the system allowing the ULA to
leverage a larger amount of spatial diversity. Furthermore, we again
observe that a stronger specular component tends to decrease the ergodic
sum spectral efficiency. The derived approximations are robust to the
presence of equal and unequal correlation matrices, as well as changes in
the level of LoS.
We also evaluate the accuracy of the limiting expected SINR expression
derived in \eqref{theorem2}, with growing numbers of BS antennas and a
fixed number of terminals in the system at $L=32$. Three trends can be
observed: After recognizing that increasing $M$ increases the expected
SINR, for each case the expected SINR slowly
saturates with growing $M$ and approaches its limiting value at
approximately 500 antenna elements for each case, respectively. This is a
result of channels from multiple terminals becoming asymptotically
orthogonal. Secondly, decreasing the physical size of the array further
reduces the inter-element spacing translating into a reduction in the
expected SINR for all cases respectively. Finally, we can observe
that each case converges to the derived limiting value.

\vspace{-5pt}
\section{Conclusion}
\label{Conclusion}
\vspace{-3pt}
In this paper, we investigated the uplink performance of large MU-MIMO
systems under spatially correlated Ricean fading,
with ULAs at the BS employed in a fixed physical space. Closed-form
approximations to the expected per-terminal SINR and ergodic sum spectral
efficiency are derived with MRC processing at the BS. In the limit of a
large number of BS antennas, asymptotic expressions for the expected
per-terminal SINR
and ergodic sum spectral efficiency were derived. Our numerical results
show that with constraints on the physical size of the ULA, the expected
SINR saturated with increasing SNR and BS antenna numbers. The analysis
accommodates to changes in system dimensions, operating SNR, LoS levels,
spatial correlation levels and variation in fixed physical spacings.
Unequal correlation matrices to each terminal resulted in a performance
increase, whilst LoS had an adverse impact on system performance.

\vspace{-5pt}
\appendices
\section{Proof of Lemma 1}
\label{ProofLemma1}
\vspace{-2pt}
We begin by recognizing
$\delta_{l}=\mathbb{E}\left[||\bm{g}_{l}||^{4}\right]=
\mathbb{E}\big[\hspace{-2pt}
\left(||\bm{g}_{l}||^{2}\right)^{2}\big]$.
Substituting the definition of $\bm{g}_{l}$ and
denoting $\bm{v}_{l}=\eta'_{l}\bm{A}_{l}\bm{h}_{l}$ and
$\bm{q}_{l}=\bar{\eta}_{l}\bar{\bm{h}}_{l}$ allows us to
state
\vspace{-3pt}
\begin{equation}
 \label{prooflemma11}
 \delta_{l}\hspace{-2pt}=\hspace{-2pt}
\mathbb{E}\big[\hspace{-2pt}\left(||\bm{g}_{l}||^{2}\right)^{2}\big]
\hspace{-2pt}=\hspace{-2pt}\mathbb{E}\big[\hspace{-2pt}
 \left(\bm{v}_{l}^{\textrm{H}}\bm{v}_{l}\hspace{-2pt}+\hspace{-2pt}
 \bm{v}_{l}^{\textrm{H}}\bm{q}_{l}\hspace{-2pt}+\hspace{-2pt}
 \bm{q}_{l}^{\textrm{H}}\bm{v}_{l}\hspace{-2pt}+\hspace{-2pt}
 \bm{q}_{l}^{\textrm{H}}\bm{q}_{l}
 \right)^{2}\big].
 \vspace{-6pt}
\end{equation}
Expanding \eqref{prooflemma11} and simplifying allows us to state
\vspace{-4pt}
\begin{equation}
 \nonumber
 \hspace{-4pt}
  \delta_{l}=\mathbb{E}\left[\left(||\bm{g}_{l}||^{2}\right)^{2}\right]=
  \mathbb{E}\big[\hspace{-2pt}\left(\bm{v}_{l}^{\textrm{H}}
  \bm{v}_{l}\right)^{2}\big]
  +\mathbb{E}\big[2\left(\bm{v}_{l}^{\textrm{H}}\bm{v}_{l}\right)
  \left(\bm{q}_{l}^{\textrm{H}}\bm{q}_{l}\right)\big]+
  \vspace{-6pt}
\end{equation}
\begin{equation}
  \label{proof1lemma12}
  \hspace{-63pt}
  \mathbb{E}\left[\bm{v}_{l}^{\textrm{H}}\bm{q}_{l}\bm{q}_{l}^{\textrm{H}}
  \bm{v}_{l}\right]+\mathbb{E}\big[\bm{q}_{l}^{\textrm{H}}\bm{v}_{l}
  \bm{v}_{l}^{\textrm{H}}\bm{q}_{l}\big]+
  \mathbb{E}\big[\big(\bm{q}_{l}^{\textrm{H}}\bm{q}_{l}\big)^{2}\big].
  \vspace{-3pt}
\end{equation}
Performing the expectation over $\bm{v}_{l}$ in the last four
terms of \eqref{proof1lemma12} and simplifying yields
\vspace{-3pt}
\begin{equation}
\nonumber
  \delta_{l}=\mathbb{E}\left[\left(||\bm{g}_{l}||^{2}\right)^{2}\right]\hspace{-3pt}=
\hspace{-2pt}\mathbb{E}\big[\left(\bm{v}_{l}^{\textrm{H}}
  \bm{v}_{l}\right)^{2}\big]+
2\left(\bm{q}_{l}^{\textrm{H}}\bm{q}_{l}\right)
PM{\left(\eta'_{l}\right)}^{2}+2{\left(\eta'_{l}\right)}^{2}
\vspace{-3pt}
\end{equation}
\begin{equation}
\label{proof1lemma13}
\hspace{-140pt}
 \bm{q}_{l}^{\textrm{H}}\bm{A}_{l}\bm{A}_{l}^{\textrm{H}}\bm{q}_{l}
+\left(\bm{q}_{l}^{\textrm{H}}\bm{q}_{l}\bm{q}_{l}^{\textrm{H}}
\bm{q}_{l}\right).
\vspace{-5pt}
\end{equation}
After recognizing that $\mathbb{E}\big[\hspace{-2pt}
\left(\bm{v}_{l}^{\textrm{H}}\bm{v}_{l}\right)^{2}\hspace{-2pt}\big]\hspace{-3pt}
=\hspace{-3pt}\mathbb{E}\left[\bm{v}_{l}^{\textrm{H}}\bm{v}_{l}\bm{v}_{l}^{\textrm{H}}
\bm{v}_{l}\right]$, substituting the definition of $\bm{v}_{l}$ and extracting the relevant
constants allows us to write
\vspace{-4pt}
\begin{equation}
\label{proof1lemma14}
\mathbb{E}\big[\left(
\bm{v}_{l}^{\textrm{H}}\bm{v}_{l}\right)^{2}
\hspace{-2pt}\big]\hspace{-1pt}
=\left(\eta'_{l}\right)^{\hspace{-2pt}4}\mathbb{E}\hspace{-2pt}\left[\left(
\bm{h}_{l}^{\textrm{H}}\bm{\Theta}\bm{h}_{l}\right)^{2}\right]\hspace{-3pt},
\vspace{-3pt}
\end{equation}
where $\bm{\Theta}=\bm{\Psi}^{\textrm{H}}\bm{\Gamma}\bm{\Psi}$ is an eigenvalue decomposition of
$\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}$. As a result,
\vspace{3pt}
\begin{equation}
\nonumber
\hspace{-51pt}
\mathbb{E}\left[\left(\bm{v}_{l}^{\textrm{H}}\bm{v}_{l}\right)^{2}\right]=
{\left(\eta'_{l}\right)}^{4}\hspace{2pt}\mathbb{E}\hspace{-2pt}\left[\left(\bm{h}_{l}^{\textrm{H}}
\bm{\Gamma}\bm{h}_{l}\right)^{2}\right]
\vspace{-5pt}
\end{equation}
\begin{equation}
\label{proof1lemma15}
\hspace{82pt}
={\left(\eta'_{l}\right)}^{4}\hspace{1pt}\mathbb{E}\left[\hspace{-1pt}\left(\sum\limits_{p=1}^{P}
\left[\bm{\Gamma}\right]_{p,p}|\bm{h}_{l;p}|^{2}\right)^{\hspace{-4pt}2}\hspace{2pt}\right],
\vspace{-3pt}
\end{equation}
where $\bm{h}_{l;p}$ denotes the $p$-th element of $\bm{h}_{l}$.
Performing the expectation with respect to $\bm{h}_{l}$ and further
simplifying yields
\vspace{-3pt}
\begin{equation}
\label{proof1lemma16}
\mathbb{E}\left[\left(\bm{v}_{l}^{\textrm{H}}\bm{v}_{l}\right)^{2}\right]
={\left(\eta'_{l}\right)}^{4}\hspace{2pt}\left\{
\left(\textrm{tr}\left[\bm{\Theta}\right]\right)^{2}+\textrm{tr}
\left[\bm{\Theta}^2\right]\right\}.
\vspace{-5pt}
\end{equation}
This allows us to write
\vspace{-2pt}
\begin{equation}
\label{proof1lemma17}
\mathbb{E}\left[\left(\bm{v}_{l}^{\textrm{H}}\bm{v}_{l}\right)^{2}\right]
\hspace{-1pt}=\hspace{-1pt}{\left(\eta'_{l}\right)}^{4}\left\{
\left(\textrm{tr}\left[
\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}\right]\right)^{2}\hspace{-1pt}+
\hspace{-1pt}\textrm{tr}\left[\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}
\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}\right]\right\}\hspace{-1pt}.
\vspace{-5pt}
\end{equation}
Recognizing that $\textrm{tr}\left[\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}\right]=PM$
allows us to state
\vspace{-2pt}
\begin{equation}
\label{proof1lemma18}
\mathbb{E}\left[\left(\bm{v}_{l}^{\textrm{H}}\bm{v}_{l}\right)^{2}\right]=
{\left(\eta'_{l}\right)}^{4}\left\{P^{2}M^{2}+\textrm{tr}\left[\left(\bm{A}_{l}^{\textrm{H}}
\bm{A}_{l}\right)^{2}\right]\right\}.
\end{equation}
Substituting the definition of $\bm{q}_{l}$ back, recognizing that
$\mathbb{E}\left[\bar{\bm{h}}_{l}^{\textrm{H}}\bm{h}_{l}\right]=M$,
combining \eqref{proof1lemma18} with the remaining terms in \eqref{proof1lemma13}
and extracting the relevant constants results in the desired expression.

\vspace{-9pt}
\section{Proof of Lemma 2}
\label{ProofLemma2}
Applying the definition of $\bm{g}_{l}$ and $\bm{g}_{k}$ into
$\mathbb{E}\left[|\bm{g}_{l}^{\textrm{H}}\bm{g}_{k}|^{2}\right]$ and denoting
$\bm{v}_{l}=\eta'_{l}\bm{A}_{l}\bm{h}_{l}$ and $\bm{q}_{l}=\bar{\eta}_{l}\bar{\bm{h}}_{l}$
yields
\begin{equation}
\label{proof2lemma21}
\varphi_{l}=\mathbb{E}\left[|\bm{g}_{l}^{\textrm{H}}\bm{g}_{k}|^{2}\right]
\hspace{-3pt}=\hspace{-3pt}\mathbb{E}\left[\big|\hspace{-2pt}\left(\bm{v}_{l}^{\textrm{H}}
\hspace{-2pt}+\hspace{-2pt}\bm{q}_{l}^{\textrm{H}}\right)
\left(\bm{v}_{k}\hspace{-2pt}+\hspace{-2pt}\bm{q}_{k}\right)\big|^{2}\right].
\vspace{-5pt}
\end{equation}
Expanding and simplifying \eqref{proof2lemma21} allows us to state
\vspace{-2pt}
\begin{equation}
\nonumber
\varphi_{l}=\mathbb{E}\left[|\bm{g}_{l}^{\textrm{H}}\bm{g}_{k}|^{2}\right]=
\mathbb{E}\left[\left(\bm{v}_{l}^{\textrm{H}}\bm{v}_{k}+\bm{v}_{l}^{\textrm{H}}
\bm{q}_{k}+\bm{q}_{l}^{\textrm{H}}\bm{v}_{k}+\bm{q}_{l}^{\textrm{H}}\bm{q}_{k}
\right)\right.
\vspace{-2pt}
\end{equation}
\begin{equation}
\label{proof2lemma22}
\hspace{-105pt}
\left.\left(\bm{v}_{k}^{\textrm{H}}\bm{v}_{l}+\bm{q}_{k}^{\textrm{H}}\bm{v}_{l}
+\bm{v}_{k}^{\textrm{H}}\bm{q}_{l}+\bm{q}_{k}^{\textrm{H}}\bm{q}_{l}\right)\right].
\vspace{-2pt}
\end{equation}
Further expanding and simplifying yields
\vspace{-5pt}
\begin{equation}
\nonumber
\varphi_{l}=\mathbb{E}\left[|\bm{g}_{l}^{\textrm{H}}\bm{g}_{k}|^{2}\right]=
\mathbb{E}\left[\bm{v}_{l}^{\textrm{H}}\bm{v}_{k}\bm{v}_{k}^{\textrm{H}}\bm{v}_{l}\right]
+\mathbb{E}\left[\bm{v}_{l}^{\textrm{H}}\bm{q}_{k}\bm{q}_{k}^{\textrm{H}}\bm{v}_{l}\right]+
\vspace{-1pt}
\end{equation}
\begin{equation}
\label{proof2lemma23}
\hspace{-95pt}
\mathbb{E}\left[\bm{q}_{l}^{\textrm{H}}\bm{v}_{k}\bm{v}_{k}^{\textrm{H}}\bm{q}_{l}\right]+
\mathbb{E}\left[\bm{q}_{l}^{\textrm{H}}\bm{q}_{k}\bm{q}_{k}^{\textrm{H}}\bm{q}_{l}\right].
\vspace{-3pt}
\end{equation}
Invoking the independence between $\bm{v}_{l}$ and $\bm{v}_{k}$, recognizing
that $\mathbb{E}\left[\bm{v}_{l}\bm{v}_{l}^{\textrm{H}}\right]
=\left(\eta'_{l}\right)^{2}\textrm{tr}\left[\bm{A}_{l}^{\textrm{H}}\bm{A}_{l}\right]$,
upon substituting back the definitions of $\bm{v}_{k}$ and $\bm{q}_{k}$ and extracting the
relevant constants, we can state
\vspace{-4pt}
\begin{equation}
\nonumber
\varphi_{l}=\mathbb{E}\left[|\bm{g}_{l}^{\textrm{H}}\bm{g}_{k}|^{2}\right]
\hspace{-3pt}=\hspace{-3pt}
\left(\eta'_{l}\right)^{\hspace{-1pt}2}\hspace{-2pt}\left(\eta'_{k}\right)^{\hspace{-1pt}2}
\mathbb{E}\left[\bm{h}_{l}^{\textrm{H}}\hspace{-2pt}\bm{A}_{l}^{\textrm{H}}
\hspace{-2pt}\bm{A}_{k}\hspace{-2pt}\bm{A}_{k}^{\textrm{H}}\right]\hspace{-2pt}+
\hspace{-2pt}\left(\eta'_{l}\right)^{2}\hspace{-2pt}\left(\bar{\eta}_{k}\right)^{2}
\left[\bar{\bm{h}}_{k}^{\textrm{H}}\hspace{-2pt}\bm{A}_{l}\right.
\vspace{-3pt}
\end{equation}
\begin{equation}
\label{proof2lemma24}
\hspace{-3pt}
\left.
\hspace{-1pt}\bm{A}_{l}^{\textrm{H}}\hspace{-1pt}
\bar{\bm{h}}_{k}\right]\hspace{-3pt}+\hspace{-3pt}\left(\bar{\eta}_{l}\right)^{2}
\hspace{-3pt}\left(\eta'_{k}\right)^{2}\hspace{-2pt}
\left[\bar{\bm{h}}_{l}^{\textrm{H}}\hspace{-1pt}\bm{A}_{k}\hspace{-1pt}
\bm{A}_{k}^{\textrm{H}}\hspace{-1pt}
\bar{\bm{h}}_{l}\right]\hspace{-3pt}+\hspace{-3pt}
\left(\bar{\eta}_{l}\right)^{2}\hspace{-2pt}
\left(\bar{\eta}_{k}\right)^{2}\left[|\bar{\bm{h}}_{l}^{\textrm{H}}
\hspace{-1pt}\bar{\bm{h}}_{k}|^{2}\right].
\vspace{-3pt}
\end{equation}
Taking the trace and simplifying yields
the result in \eqref{lemma2}.

\vspace{-10pt}
\section*{Acknowledgment}
\label{Acknowledgment}
The authors would like to thank Prof. Andreas F. Molisch at
the University of Southern California for the insightful
discussions during the course of this work.

\vspace{-4pt}
\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
Late Night TV
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Carly Fiorina sang a tune she'd concocted for her dog, talked about the importance of faith and defended her track record at Hewlett-Packard to Jimmy Fallon on The Tonight Show. In 2015, displays of personality by presidential candidates still are considered so daring that her performance set off a flood of press reports comparing her performance to White House hopeful Bill Clinton playing the sax on Arsenio Hall's late-night show nearly a quarter-century ago.
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She told her story of meeting Vladimir Putin and got to repeat her "no talking" stance with regard to the Russian leader that she spelled out during the most recent GOP debate, which wowed the crowd and led most pundits to declare her the winner. On Fallon's show, she went one step further and compared Putin to Trump:
https://youtu.be/bhdkBFwCWtw
8 David Straiton & Nora Lonergan To Produce Series Based On Story Of African Rock Band Wells Fargo For The Africa Channel Studios; Daisy Martey Tapped To Write Pilot
10 Will Smith, Jennifer Hudson, 'The Harder They Fall' Among Winners At AAFCA Awards
'Sunday Night Football's Steelers-Chiefs Game Earns More Than 20M Viewers & Tops Primetime; '60 Minutes' Leads Non-Sports


as it roared off down the track.
but we'll catch it, up on the mountain's back.
and hold it at their fence.
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the challenge is there for us all.
We can take it, do battle and lose.
Or we may fight, and win after all.
we'll depth our spirit in the firey din.
the fire will gain while they do.
see nought, but victory in view.
and can live with the truth we find.


text:
Aloe mahraensis är en grästrädsväxtart som beskrevs av John Jacob Lavranos och T.A.Mccoy. Aloe mahraensis ingår i släktet Aloe och familjen grästrädsväxter.

Artens utbredningsområde är Sydjemen. Inga underarter finns listade i Catalogue of Life.

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mahraensis

redpajama_set_name: RedPajamaWikipedia


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text:
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BlogHome » NEWS » Uncategorized » mangrove swamp climate graph
mangrove swamp climate graph
Mangrove ecosystems should be better protected, the UN's food agency has warned as it published new figures showing that 20% of the world's mangrove area has been destroyed since 1980 More information about general climate impacts to habitats in Florida. Lernen Sie die Übersetzung für 'mangrove swamps' in LEOs Englisch ⇔ Deutsch Wörterbuch. Mangroves: 11 facts you need to know These unique trees lead tough lives — but we're all the better for it. Buttonwoods usually occur in areas above high tide. Changes in freshwater input could alter the species composition of the mangrove habitat.More information about general climate impacts to species in Florida. Loss of mangrove habitat will impact these species' reproductive success and survival. If other conditions are suitable, these bird species may be able to expand their range northward along with the mangroves. The temperatures or water depths in the following section are sometimes taken from specific places, which are marked next to them for further understanding. In the Nanliu River Estuary of Guangxi, China, the naturally expanding process of a mangrove swamp (primarily consist of Aegiceras corniculatum) over past decades is studied by satellite images. Australian mangrove forests comprise 45 plant species from 18 families, which is more than half the world's mangrove species. Mangroves provide nesting habitat for many bird species, including wading birds, black-whiskered vireo, mangrove cuckoo, and Florida prairie warbler. The expanding process is not gradual and the significant expansion only took place in some … Weather/Climate The weather in this biome should consits of lots of precipitation. Mangroves are tropical trees that thrive in conditions most timber could never tolerate — salty, coastal waters, and the interminable ebb and flow of the tide. Often vines, such as rubber vines and morning-glory, climb over mangroves, especially at swamp edges.This conservation asset includes Mangrove Swamp, Scrub Mangrove and Buttonwood Forest. Mangrove swamps occur in warm climates near the _____. Mangroves have evolved to survive in coastal wetland environments where the salt and oxygen poor soil makes it impossible for other plants to live. Mangroves are an important protection against climate change, with one acre (4,000 sq m) of mangrove forest absorbing nearly the same amount of carbon dioxide as an acre of Amazon rainforest. What are mangroves? About Climate statistics | Data file of statistics for this site (csv) | Site selection menu. In fact, 75% of all of the Mangrove Forests exist between 25º North and 25º South. Future threats to mangroves include impacts from climate change, such as increased air and water temperatures, levels of atmospheric carbon dioxide (CO2), and sea level rise. Climate Change May Wipe Out Large Mangrove Forests, New Research Suggests Mangroves help protect coastal areas from flooding and sequester more carbon than … Almost 100% of the current extent of mangroves is expected to be inundated by 1m of sea level rise; however, it is expected that mangroves may be able to expand their range inland and possibly northward as temperatures increase and cold weather events decrease. In coastal areas with low elevation gradients mangroves are likely to be overwhelmed by a rapid rate of sea level rise. Globally, mangrove habitats cover a total area of around 52,300 square miles, and exist on every continent that has subtropical and tropical climate zones. Purchase upland development rights or property rights. More than one in six mangrove species worldwide are in danger of extinction as a result of a number of factors, including coastal development, climate change, logging and agriculture. Large expanses of mangroves could be fragmented into smaller, more isolated patches, as mangroves in some areas are able to keep pace and/or migrate while those in other areas won't be able to migrate. These swamp communities are usually composed of red mangrove, black mangrove, and white mangrove. The "beautiful forests" or the Sundarbans, home to the swamp tigers, wears a crown of many jewels. Extremes graphs – monthly; Extremes graphs – annual; Climate records; About Australian climate Climate statistics for Australian locations Monthly climate statistics All years of record. Protect buffer zones landward of shore to allow for future upslope migration. One tree species, Avicennia integra, is found only in Australia - in the Northern Territory, east of Darwin. Papua New Guinea - Papua New Guinea - Climate: Although all the climatic regions of Papua New Guinea are basically tropical, they are nevertheless varied. Most of the mangrove forestation is in the Indian Ocean, whether it be on the coasts of India or surrounding the islands of Indonesia (see Locations of Forests).The climate in this area of the world varies from day to day, with the yearly average being 22 °C (72 °F). Time in the sun. However, they can keep pace with more modest rates of sea level rise, moving inland as the seas rise. Report on Visit to Mangrove Swamp: Draft a short report about visit to the mangrove swamp. Preserve and restore the structural complexity and biodiversity of vegetation. More information about adaptation strategies. That said, one comprehensive study of mangroves notes that climate change may prove to be the ultimate anthropogenic disturbance factor, resulting in a maximum loss of 10 15% of – mangrove ecosystems globally (Alongi, 2008). These forests are comprised of 12 genera comprising about 60 species of salt-tolerant trees. Protect critical areas that are naturally positioned to be more resistant to climate change. From 1988 to 2013, the area of studied mangrove swamp increased significantly from 60 hm2 to 134 hm2. Each mangrove tree species is specific to particular latitudes and levels of tidal inundation. Mangrove Forests exist in many different locations around the world, but are mostly in the same range of latitudes. More information about climate change interactions with existing threats and stressors in Florida. Implement best management practices to reduce sources of land-based pollutant and nutrient loads impacting species and habitats. Mostly because that keeps the swamps actually wet and also to help the plants and animals that live their because they need lots of water to remain alive. Weather in Mangrove Forests can ranges drastically because of their location in mainly tropical climates around the world. Especially in the tropical Mangrove Forests, there are often extreme weather patterns, like tsunamis or hurricanes. Equal quantities of water. Eliminate/reduce mangrove thinning/pruning. Of these impacts, rise in sea‐level is expected to be the greatest factor influencing the Protect mangrove areas that demonstrate persistence over time – areas with a range of small young and large old trees or mangrove roots with dense epibiont communities such as oysters, sponges, tunicates and corals. By the end of this century, less frequent and less intense freeze events are expected to enable the poleward migration of mangroves. In addition to mangrove trees, mangrove habitats are also home to a number of other plant species. These swamp communities are usually composed of red mangrove, black mangrove, and white mangrove. Mangroves occur in the waterlogged, salty soils of sheltered tropical and subtropical shores. Educate planners on the importance of coastal habitat preservation (including serving as hazard buffers), climate change and incorporation into long-range planning efforts. Work with volunteers to control invasive species. Actively engage with communities to minimize urban encroachment. Mangroves form dense, brackish-water swamps along low-energy shorelines and in protected, tidally influenced bays of southern Florida. Impacts of Climate Change on Mangrove Ecosystems Sea level rise Processes contributing to mangrove vulnerability to sea level rise. The beautiful mangrove forests of the Sundarbans. Enhance outreach regarding impacts of fertilizer on water quality. Protect mangrove forests with abundant mature trees producing a healthy supply of seeds and propagules. Subspecies A. m. marina is found in Western Australia from Bunbury in th… They stretch from the intertidal zone up to the high-tide mark. Restore hydrology by filling mosquito ditches, removing roads, and replacing culverts with types/sizes that will accommodate future flows. Since 2015, Oemar Idoe has been the Principal Advisor of the "Management of the Sundarbans Mangrove Forest for Biodiversity Conservation and Increased Adaptation to Climate Change Project" (SMP) at the Deutsche Gesellschaft für Internationale Zusammenarbeit (GIZ) in Dhaka, Bangladesh. Image credit: Dr. Anish Andheria . Standardized Index of Vulnerability and Value Assessment - Natural Communities. Up until two decades ago, scholars have focused on the provisioning, supporting, and cultural services associated with mangroves, but recent efforts have pivoted towards mangroves role in regulating the global carbon cycle. Mangroves are marine trees that form coastal wetlands known as mangrove forests and can be found around the world in tropical and sub-tropical climates. Dam operations/incompatible release of water, Incompatible wildlife and fisheries management strategies. What are synonyms for mangrove swamps? This conservation asset was assessed as part of the Standardized Index of Vulnerability and Value Assessment - Natural Communities (SIVVA).This conservation asset has a SIVVA vulnerability score greater than 70 but is not among the top 5 most vulnerable natural communities in any SIVVA vulnerability category.Read more information about SIVVA natural communities. They are often in a tropical setting, near the equator. The Impact of Ecosystems: Mangrove Wetlands Stabilize Climate Change. And the climate keeps on changing, bringing ever more erratic rains, storms, and temperature swings. Mangrove Swamps Climate Mangrove swamps occur along tropical and subtropical coastlines. answer choices . Mangroves in the context of a changing climate . Mangrove Mt climatology including monthly and daily averages and records of temperature and rain. What is the outcome (dependent) variable in this experiment? Areas where mangroves occur include estuaries and marine shorelines.. More information about general climate impacts to species in Florida. In addition to moving inland, mangroves are likely to continue a northward range expansion. Develop coordinated foreshore habitat management plans that establish a connected network of protected areas across the entire Florida shoreline. Mangroves are commonly found along sheltered coastlines in the tropics and sub-tropics where they fulfil important socio-economic and environmental functions. Include observations, graph to show quadrants along line transect and relationship between height of water and the quadrants along the line transect; describe connections between plants/animals; and summarize learning from classroom discussions. The existence of mangrove peat deposits worldwide attests to past episodes of local and regional extinction, primarily in response to abrupt, rapid rises in sea level. Replace hardened shorelines with natural vegetated shorelines. Mangroves form dense, brackish-water swamps along low-energy shorelines and in protected, tidally influenced bays of southern Florida. They may be many areas where inland migration is prevented due to barriers/human development. Climate change is expected to alter the frequency and intensity of climate extremes, which will greatly affect mangrove forests. 1 synonym for mangrove: Rhizophora mangle. Dark color vs. light color. Mumbai's Disappearing Mangroves: A Weapon Against Climate Change Is Under Threat : Goats and Soda Studies show that the city lost nearly 40 percent of its mangroves … 22 °C (72 °F). Establish protections for transitional habitats that will provide for range shifts and serve as potential climate refugia. Relatively small changes in winter temperature (e.g., increasing minimum temperature) can result in dramatic mangrove range expansion. He constructed a graph to display the average temperature of each can over the course of his experiment. Depending on slopes and amounts of disturbance, mangrove swamps may progress in zones of single species from seaward (red mangrove) to landward (white mangrove) areas. Work with counties, local municipalities and regional planning councils to incorporate natural resources adaptation strategies in comprehensive plans and hazard planning efforts. Restore mangrove communities that are most likely to withstand climate change impacts if other threats are removed (non-climate stressors). Create your own unique website with customizable templates. Mit Flexionstabellen der verschiedenen Fälle und Zeiten Aussprache und … Work with communities and landowners to choose vegetation, living shorelines, oyster reef restoration, or hybrid approaches in favor of traditional hard armoring. This is largely because of the physiography and climate of the west coast. Mangroves are distributed in the land‐sea interphase of tropical and subtropical coasts in 123 countries. Work with communities to reduce stormwater runoff. A total of 0.4 million hectares (48 per cent) are found in Queensland and 0.3 million hectares (37 per cent) are in the Northern Territory. Open a high-resolution version of Map 1 that can be saved as a PNG file The most common mangrove is white mangrove (Avicennia marina), which is found in mainland coastal areas and comprises three subspecies. The intertidal existence to which these trees are adapted represents the major limitation to the number of species able to thrive in their habitat. Antonyms for mangrove swamps. This community type is composed of freeze-sensitive tree species and, with some limited exceptions, mangroves which are distributed south of Cedar Key on the Gulf coast and south of St. Augustine on the Atlantic coast. The Unique Address Of The Swamp Tigers. In these areas, there are usually relatively high temperatures (70-80 degrees Fahrenheit) and significant amounts of precipitation (4.5 meters). We have already lost over half of the world's original mangrove forest area, estimated at … This community type is composed of freeze-sensitive tree species and, with some limited exceptions, mangroves which are distributed south of Cedar Key on the … Summary statistics MANGROVE MOUNTAIN AWS. Animals in the Swamps Read more information about SIVVA natural communities. These include the provision of a large variety of wood and non-wood forest products; coastal protection against the effects of wind, waves and water currents; conservation of biological diversity – including a number of endangered mammals, reptiles, amphibians and birds; protection of coral reefs, sea grass beds and shipping lanes against siltation; an… Mangroves are well studied the world over. Here's what that means for us. Mangrove Forests are very helpful at times such as these because they prevent the erosion of coastlines with their anaerobic roots supporting the sandbars. They are subject to the twice-daily ebb and flow of tides, fortnightly spring and neap tides, and seasonal weather fluctuations. Synonyms for mangrove swamps in Free Thesaurus. 10th - 12th grade. Increased fragmentation of the mangrove habitat could impact wading birds that nest in large colonies, with smaller more isolated colonies potentially less successful.
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Council has primary responsibility to coordinate and manage disasters and emergency events if they occur in the region. This is stated under the Disaster Management Act 2003. Council receives support from the State government.
lead and enable local recovery.
to preserve lives, livelihoods and the environment.
The safeguarding of human life is priority. The Local Disaster Management Group (LDMG) reviews the plan annually.
Council established the LDMG in May 2008. It is part of Queensland's broader disaster management system. This allows for a gradual increase of support at a local, district and state level as required.
to perform a function incidental to a function mentioned above.
The LDMG meets four times each year and the meetings are timed to take into account higher risk periods.
coordinates resources to support local governments.
It is chaired by a senior police officer (the District Disaster Coordinator) who has wide ranging powers when a disaster is declared.
plans for the allocation and coordination of those resources.
The district group works closely with the local groups to support them.


Can you volunteer for one of these nightly duties per week for the 12-week program?
Intake Center: United Methodist Church: 5 to 6:30 p.m.
Host Site Welcome Desk/Sleeping Bag Desk: 5:30 to 8:30 p.m.
Greet dinner/hospitality volunteers-sign in when you arrive at 5:30 p.m.
Check sleeping bags and cots when you arrive at 5:30 p.m.
Greet guests-assign chores at 6 p.m.
Give OTC medications to guests-keep track of info.
Eat dinner with guests at 7 p.m.
Prepare evening report before leaving at 8:30 p.m.
Evening Drivers: 5:45 to 6:30 p.m.
Evening drivers arrive at Intake Center (Elk Grove United Methodist Church) at 5:45 p.m. and transport guests to Host Site.
Morning Drivers: 6:30 to 7 a.m.
Morning drivers arrive at Host Site at 6:30 a.m. and transport guests back to Intake Center (Elk Grove United Methodist Church).


Private jet operator enhances aircraft with connectivity management services to better control in-flight connectivity costs and usage.
SHANGHAI, April 16, 2019 — Honeywell's (NYSE: HON) has been selected by Hong Kong-based HK Bellawings Jet to supply GoDirect™ Cabin Connectivity applications for the company's growing aircraft fleet. With Honeywell's GoDirect Cabin, HK Bellawings will have access to cost-efficient Connected Aircraft services to more effectively manage cabin connectivity services.
HK Bellawings Jet will use GoDirect Cabin to help manage passenger data consumption during flight to better control overhead costs associated with in-flight connectivity. By using GoDirect Data Control, a service included with GoDirect Cabin, operators can save up to $100,000 per year, per aircraft, on their data costs depending on the activity of their passengers. In addition, HK Bellawings will also use GoDirect Access voucher cards to manage in-flight internet connections and data usage. The voucher card can provide passengers with a fixed amount of data and lets them purchase more, if needed.
GoDirect Cabin offers a comprehensive suite of scalable satellite connectivity services to meet nearly every need. This includes access to GoDirect apps and all the benefits of reliable in-flight connectivity thanks to its next-generation routers and routing software. These services give operators the flexibility and visibility into their current wireless services to make troubleshooting, networking and onboard data management an easy process.
To learn more about Honeywell's GoDirect connectivity hardware, apps and services, visit Honeywell at ABACE 2019, booth B435 in Hall B, Shanghai, China, April 16–18.
GoDirect Cabin is part of Honeywell's GoDirect family — the industry's most complete portfolio of more than 50 value-adding aviation services and applications that provide operators, flight crews and maintenance teams with critical information to control their services and network.


It's back to school for most children in America, but tooth pain can be a real distraction from important studies. Certain dental aches and pains are inevitable, harmless, and pass quickly for kids. Babies and toddlers naturally experience some discomfort as their teeth form in their gums and break through the surface. As children get older and lose primary pearly whites, their permanent teeth emerge and may cause a slight ache here and there. All of this is a normal part of growth and development. However, if your child complains of sharp pain or prolonged aches in their mouth, you'll definitely want to give West Des Moines dentist, Dr. Jessica Johnson, a call.
Wearing Braces: If your child wears braces or any orthodontic appliances, they may experience discomfort, especially following a visit to the have their braces altered during treatment. Tightening of the wires or a placement of new rubber bands are usually the main causes of aches. If the pain subsides in a few days, then you know it's normal. However, if the pain continues or becomes worse, contact your dentist or orthodontist to check for other underlying concerns.
Bad Habits: Your child may have a physical habit causing their tooth pain. Does your daughter crunch ice or chew on a hard toy regularly? Help her to break the habit with reminders and a reward system. Does your son grind his teeth at night or clench her jaw? Experts report that two out of every three kids grind or clench their teeth at night. Dental professionals call this condition bruxism. Observe your children while they are sleeping. If you hear grinding and your child is complaining of tooth pain while chewing, make an appointment with Dr. Johnson so she can check for wear on your child's teeth. Treatment can be as simple as a nightguard to halt the grinding.
If you have had a recent filling, you may have been perplexed if your dentist told you to bite down on a colored piece of paper, and then asked: "does your filling feel too high?" Most patients experience a degree of anxiety about having a dental procedure performed on them, naturally, so when you've come to the end, you're likely ready to go home. Add this to the challenge that your mouth is probably still numb from general anesthetic, and you may find the high filling question a little difficult to answer. However, if you end up with a filling set too high, you're likely to end up feeling considerable discomfort.
Why Does a High Filling Cause Discomfort?
High fillings put stress on your periodontal ligament. The inflammation causes tenderness, discomfort, and sometimes out-and-out pain. Every time you bite down with a high filling, you can even end up with acute apical periodontitis (also known as symptomatic apical periodontitis). Feeling pain at the site of your new filling might really concern you. Could this be an infection? Am I going to have to get the filling done all over again? Luckily, if your problem is a high filling, your dentist will likely have a swift solution. Going in to see Dr. Johnson to get the high filling ground down to the right level to even out your bite will usually only require a few minutes back in the dental chair.
After you've left your dentist's office, and the local anesthetic has worn off, a little discomfort might be usual with a new filling. However, if you feel actual pain, or the discomfort lasts for days, you likely have a high filling. You may also feel a misalignment when you bite down. If your new filling just doesn't feel right, call your dentist immediately. Once the filling is corrected, your periodontal ligament may still need a little time to heal, but you are likely to feel as good as new within two weeks.
Every six months, you should visit Des Moines dentist, Dr. Jessica Johnson for a checkup and cleaning. This is the most effective way to avoid severe gum disease and cavities which can heighten your risk for tooth decay, dental abscesses, tooth loss, and jawbone depletion. Once your oral health becomes that declined, it can be costly (both time wise and financially) to get your grin back in shape again. Preventive cleanings are the type that you're used to during your biannual dental examination schedule. If you get off track or avoid going to the dentist, you may require a deeper cleaning.
The typical dental cleaning is also clinically known as prophylaxis. This is the sort of dental examination which most people receive when they attend a dental visit on schedule. The focus of this type of cleaning is maintenance; namely plaque/tartar removal. Preventive dental checkup steps help to prevent oral infections, such as periodontal (gum) disease and cavities. When performing a basic dental checkup, a dental hygienist may use a variety of tools to scrape plaque and tartar buildup from your teeth. The hygienist will also floss teeth to remove bacteria from the surfaces that are unreachable by dental instruments, and polish your teeth with a powered toothbrush.
A second type of dental cleaning is necessary when bacteria are winning the battle inside your mouth. Also known as root planing and scaling, deep cleanings are typically performed on patients who have developed gum disease which is getting beyond gingivitis (early gum disease). A patient with periodontitis (severe gum disease) often develops bacteria-filled pockets between teeth and gums. These germs attack the base of teeth and the surrounding periodontal ligaments and tissues. If you ignore this condition, periodontitis can result in tooth loss.
When was the last time you had a dental cleaning? If it has been more than six months since your last dental visit, you are due for a checkup. Schedule an appointment with Dr. Jessica Johnson today by calling 515-223-9764. Our 50266 dentist office welcomes patients from West Des Moines, Waukee, Clive, and nearby Des Moines metro communities.
You already know that smoking is incredibly harmful to your overall health. The number of diseases and unpleasant side effects which are directly related to the unfortunate habit is staggering. Beyond the bad breath, yellowing teeth, and oral cancer risk, you can add tooth loss to the list. West Des Moines dentist, Dr. Jessica Johnson, notes a research study which shows the devastating link between oral health issues and tobacco use.
Tufts University was the site of a study which tracked the oral health of approximately 500 men for 30 years. The male participants were initially divided into two groups: smokers and non-smokers. As some of the male smokers quit the habit, they formed a third group. On average, the consistent smokers lost almost three teeth every ten years. The non-smokers lost slightly more than one tooth over the same period of time. The group of smokers who quit lost an average of 2.5 teeth while smoking. However, the rate of tooth loss declined dramatically after they quit. Tufts conducted a similar study among roughly the same amount of women. The researchers found that smoking women were two times as likely to lose teeth as non-smoking females.


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November 8, 2013 August 29, 2013 Shala Howell
Thou Shalt Not Play Sports Nor Smoke Tobacco
Reposted from Once Upon a Time in Needham.
The Great Pastoral Coup d'Etat of 1903
Cyclists in the first Tour de France in 1903. Men of dubious moral character, all. Especially that guy in the white jersey. He's Maurice Garin, aka "The Little Chimney Sweep," and had the dubious distinction of winning the race. (Image: AP Photo/Conservatoire du Patrimoine Sportif)
Every once in a while I come across something in the archives of the Congregational Church of Needham that reminds me how very different life used to be. For example, the firing of the Rev. Frank Estabrook in May 1903 after a long and contentious spiritual battle with church members.
In his blistering six-page resignation letter, Rev. Estabrook recommends that the congregation take a hard look at its by-laws for:
"only a broad and liberal platform of faith, an immediate and radical revision of your By-laws, and the utmost consecration and devotion on the part of all your members can make you even moderately successful. I see no good ground to hope for any of these things."
Several pages of the letter are devoted to describing an intolerable situation in which the Rev. Estabrook was subjected to a
"season of exaggeration whispered in dark places… and tale-bearing which it is revolting to recall."
If he were to stay on as pastor, he writes, he could
"neither forget these things nor repair the damage that has been done."
What on earth could have caused this rift?
"I frankly admit that I do not agree with many members of this church in matters of faith and personal practice. Indulgence within the limits of temperance in all sport and games I consider not only harmless but sane, wholesome and beneficial. The horror with which many of you regard the use of tobacco finds no sympathetic response in me."
That's right. The congregation forced the Reverend Frank P. Estabrook to resign because he liked to play sports and smoke tobacco.
1907: The Burning of the Mortgage (Once Upon a Time in Needham)
Congregational Church of Needham
first Tour de France
Needham Massachusetts
Rev. Frank Estabrook
smoking tobacco
Turn-of-the-century sports
Published by Shala Howell
I spent two decades helping companies like Bell Labs, Juniper Networks, and a genetic testing company that was later acquired by CVS translate some of the world's most complicated concepts into actionable, understandable English. Now I'm working on a different problem -- helping parents raise the next generation of critical thinkers. The first book in my Caterpickles Parenting Series, What's That, Mom?, focuses on how to use public art to nurture children's curiosity in the world around them. My next book will focus on science, and how parents without a science degree can answer their curious child's questions without enrolling in a college level refresher course.	View all posts by Shala Howell
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This work by Michael and Shala Howell is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.