The intersection at Baltimore Pike and Route 796 in Penn Township is arguably the worst one in southern Chester County. But the Penn Township Board of Supervisors is working with engineers on a plan for major improvements that will ease congestion and improve safety.
Developing a concept plan is a prerequisite for applying for grants to pay for a large reconstruction of the intersection. The new design will realign the intersection and add turning lanes on both Baltimore Pike and Route 796.
At a recent meeting where the intersection improvements were discussed, Curtis Mason, the chairman of the board of supervisors, said that the intersection has become dangerous because of the increased traffic in the township. Mason said that the intersection is the most pressing issue in the township, and we agree.
Penn Township has developed into a commercial hub in southern Chester County, and is the home to several retirement communities and health care facilities. It is a lot busier than it was even a decade ago, and the Baltimore Pike and Route 796 intersection is crucial to the township.
Fortunately, Penn Township is well-managed, and supervisors long ago identified the need for intersection improvements and had the foresight to take action.
The Penn Township supervisors voted to purchase the Red Rose Inn, one of the more historic buildings in the township, with the idea that intersection improvements were inevitable. It's great that the township had the ability to purchase and preserve an historic building, but it's equally important that the township owns that property in advance of reconstruction work on the intersection. The original part of the Red Rose Inn, which has historic value, will remain intact. But some of the side structures on the Red Rose Inn property will need to be removed to allow for the road to be widened.
Penn Township has one of the lowest tax rates in the area, and is historically on the conservative side on financial issues. You can bet that purchasing an historic property was not at the top of the supervisors' wish list. But that decision will pave the way for much-needed improvements to a dangerous intersection. We applaud this example of good governing by the Penn Township Board of Supervisors, which includes Mason, Victor Mantegna, Tom Barlow, Ken Bryson, and Bill Finnen.


text:
\section{Introduction and Background}

In the tradition of Gabor's 1946 landmark paper \cite{Gabor46a}, we advocate a time-frequency (TF) approach to communications.
TF methods for communications have been proposed very early (see the box \textsc{History}).
While several tutorial papers and book chapters on the topic are available
(see, e.g., \cite{wanggg00,magg06,ltvbook11} and references therein), the goal of this paper is to present the fundamental
aspects in a coherent and easily accessible manner. Specifically, we establish the role of TF methods in communications across a range of
subject areas including TF dispersive channels, orthogonal frequency division multiplexing (OFDM),
information-theoretic limits, and system identification and channel estimation. Furthermore, we present fundamental results that are stated in the literature
for the continuous-time case in simple linear algebra terms.

\begin{figure*}[tbh]
\renewcommand\baselinestretch{1.1}\small\normalsize
\begin{center}
\noindent\includegraphics[width=0.8\textwidth]{intro_pic2.pdf}
\end{center}
\vspace*{-5mm}
\caption{\label{fig:intro}Top: A communication system consisting of a transmitter, a noisy channel, and a receiver.
Bottom: Illustration of the TF shift structure of an OFDM modulator, a (measured) TF dispersive channel (see \cite{cfm_proc11}), and an OFDM demodulator.}
\vspace*{-1mm}
\end{figure*}

We consider a point-to-point communication scenario with a single transmitter, a channel, and a single receiver as shown in Fig.~\ref{fig:intro}.
The channel models the transmission medium and imperfections of transmitter and receiver hardware like oscillators, amplifiers, and antennas.

A basic element of TF analysis is the TF shift operator $\M_\nu\D_\tau$, which induces a delay (time shift) $\tau$ and a modulation
\revv{(frequency shift)}
$\nu$ according to
$\big(\M_\nu\D_\tau x)(t) = x(t-\tau)e^{j2\pi\nu t }$ \cite{groebook,fla-book2}. TF shifts are fundamental to communications in a twofold manner:

\begin{enumerate}

\item Many linear channels are TF dispersive, i.e., they induce time dispersion (delays) and frequency dispersion (modulation).
These channels can be represented as a weighted superposition of TF shift operators \cite{groebook}.

\vspace{.5mm}

\item OFDM is a multicarrier transmission scheme that modulates the data symbols onto Weyl-Heisenberg (WH) function sets,
also known as Gabor sets \cite{groebook,Fei_Str_97}. These function sets consist of TF shifted versions of a prototype pulse (Gabor's ``logons'' \cite{Gabor46a}).

\vspace{.5mm}

\end{enumerate}

\begin{floatbox}[t!]
\vspace*{1mm}
\begin{center}
\begin{minipage}[t]{0.98\columnwidth}

\begin{center}
{\sc History}
\end{center}
\vspace*{-1mm}

TF analysis has been linked with communications for a long time. Gabor \cite{Gabor46a},
father of the Gabor expansion, proposed the use of ``TF logons'' (TF shifts of a prototype pulse) to represent communication signals.
Zadeh \cite{Zad50} introduced a TF transfer function of TF dispersive systems.
Chang \cite{Chang66} proposed the multicarrier transmission scheme known as orthogonal frequency division multiplexing (OFDM).
Kailath \cite{Kai62} discussed the sampling and measurement of TF dispersive systems.
Bello \cite{Bel63} studied random TF dispersive channels and introduced the concept of wide-sense stationary uncorrelated scattering (WSSUS).
The estimation of WSSUS channel statistics was addressed by Gallager \cite{Gal64} and Gaarder \cite{Gaa68}. An extensive
discussion of communication over random TF dispersive channels was provided by Kennedy \cite{Ken69}.

\end{minipage}
\end{center}
\vspace*{-1mm}
\end{floatbox}

OFDM and TF dispersive channels are at the heart of a broad range of communication systems,  including
Digital Audio/Video Broadcasting, Wireless Local Area Networks (IEEE 802.11), Wireless Metropolitan Area Networks (IEEE 802.16),
3GPP Long-Term Evolution, Wireless Personal Area Networks (e.g., WiMedia), Vehicular Ad Hoc Networks, L-band Digital Aeronautical Communication Systems,
Digital Subscriber Lines, power-line communications, and underwater acoustic communications
\cite{cfm_proc11,ladebusch06,hanzo-mimo-ofdm,haas2002,corripio06,baoshengli08,antonia_book11}.
The purpose of this paper is to discuss the relevance of TF analysis to OFDM and TF dispersive channels, and to demonstrate that
WH frame theory
\cite{christensen03}
and TF operator representations are powerful tools for pulse design \cite{Haas96,kozmol98,boel_spie99,schniter_tsp07,Matz-Charlypaper07},
capacity analysis \cite{durisiboel_book11}, and channel identification (sounding, estimation) \cite{kozek_identification_2005,sounder_tc00,heckelboel12}.
We note that parts of this paper draw on our previous work in
\cite{GM-Hla-jmp98,boel_spie99,Matz-Charlypaper07,heckelboel12,durisiboel_book11}, and \cite{ltvbook11_ch1}.

\section{TF Dispersive Channels}

In this section, we discuss the physics, system theory, and statistics of TF dispersive channels.

\vspace*{-2mm}

\subsection{Physics}

First, we describe various physical mechanisms that entail a superposition of TF shifts.

\vspace*{1mm}

\defc{c}

\subsubsection{Multipath Propagation and Doppler Effect} In wireless (radio or underwater) communications, the electromagnetic or acoustic wave
propagating from the transmitter to the receiver may interact with objects in the environment. These objects are commonly
referred to as scatterers, even though the interaction mechanism may include reflection and diffraction. The wave usually propagates along several
distinct paths with different propagation delays and attenuation factors. This situation is known as multipath propagation.

If transmitter, receiver, or scatterers are  moving, the
Doppler effect entails a time scaling (equivalently, a frequency scaling)
of multipath components. For narrowband signals, i.e., signals whose spectrum is supported in a small band around a carrier frequency $f_\text{c}$,
frequency scaling can be approximated by a frequency shift of $\nu = v f_\text{c}/c$, where $c$ is the wave propagation speed and
$v$ is the velocity of the moving object in the direction of wave propagation.

In general, the transmitted signal $x(t)$ is affected by both multipath propagation and Doppler frequency shifts. Assuming $I$ specular paths
with delays $\tau_i$, Doppler frequencies $\nu_i$, and complex gains $S_i$, the receive signal is given by the following
weighted superposition of TF shifts of $x(t)$:\footnote{Additive noise is neglected throughout this section.}
\[
y(t)
= \sum_{i=1}^I S_i\,x(t-\tau_i)\,e^{j2\pi\nu_i t} = \sum_{i=1}^I S_i\hspace*{.1mm}\big(\M_{\nu_i}\D_{\tau_i} x)(t).
\vspace*{1mm}
\]

\subsubsection{Medium Variations} Many transmission media, such as cables and optical fibers, are characterized by
material dispersion, i.e., a group velocity that varies with frequency. Material dispersion can be modeled by a time-dispersive channel that is
described by the convolution relation\footnote{Integrals are over the entire real line.}
$y(t)= \int g(\tau)\,x(t-\tau)\,d\tau = \int g(\tau)(\D_\tau x)(t)\,d\tau$, i.e., the receive signal is a weighted superposition of time-shifted versions of the
transmit signal.

In the presence of environmental changes, switching effects, or component drift, the transmission medium varies over time.
Such variations can be modeled by a frequency-dispersive channel with multiplicative input-output relation $y(t)=m(t)x(t)$.
Denoting the Fourier transform of $m(t)$ by $M(\nu)$, the equivalent relation $y(t)= \int M(\nu)(\M_\nu x)(t)\,d\nu$ shows that the
receive signal is a weighted superposition of frequency-shifted versions of the transmit signal.
General channels may exhibit both time and frequency dispersion.

\vspace*{1mm}

\subsubsection{Oscillator Imperfections and Timing Offsets} In most communication systems, the baseband transmit signal is modulated onto a
sinusoidal carrier via an oscillator. The receive signal is then demodulated, ideally using the same sinusoidal carrier.
However, practical oscillators exhibit imperfections such as frequency offset and phase noise. Furthermore, transmitter and receiver suffer from a timing (clock) offset.
Consider, for example, a receiver with frequency offset $\Delta f$, phase noise $\phi(t)$, and timing offset $\Delta t$, and an otherwise ideal transmission medium.
Here, the baseband receive signal is given by
\[
y(t) = e^{-j2\pi f_\text{c} \Delta t}\! \int \Psi(\nu\!+\!\Delta f) (\M_\nu\D_{\Delta t} x)(t)\,d\nu ,
\]
where $\Psi(\nu)$ denotes the Fourier transform of $e^{-j2\pi \phi(t)}$. Frequency offset, phase noise, and timing offset thus amount to a superposition of TF shifts.

\vspace{-1mm}

\subsection{Elementary Channel Characterizations}

We next review system-theoretic aspects of TF dispersive channels. In what follows, frequency shifts will
be referred to as Doppler shifts even if the underlying physical mechanism is not the Doppler effect.
The basic input-output relation of a TF dispersive channel $\H$ is denoted as $y(t)=(\H x)(t)$.

\vspace*{1mm}

\subsubsection{Delay-Doppler Spreading Function}

We have seen that different physical effects amount to a weighted superposition of TF shifts. In fact, it is shown in \cite[Th.~14.3.5]{groebook}
that virtually any linear channel (operator) $\H$ can be represented as a (generally continuous) superposition of TF shift operators in the sense that
\begin{equation}
y(t) = \iint  S_{\H}(\tau,\nu)\, \big(\M_\nu\D_\tau x)(t) \,d\tau\,d\nu.
\label{eq:spreading_io}
\end{equation}
For the finite-dimensional case, a simple explanation of this representation result is given in the box {\sc Operator Representation}.
The function $S_{\H}(\tau,\nu)$ in \eqref{eq:spreading_io} characterizes the complex weight associated with delay $\tau$ and Doppler shift $\nu$ and is known
as delay-Doppler spreading function. We note that even though \eqref{eq:spreading_io} applies generally, in the (ultra)wideband regime more parsimonious
channel representations may be obtained using Doppler scaling instead of Doppler shifts \cite{antonia_book11}.

\begin{floatbox}[t!]
\vspace*{1mm}
\begin{center}
\begin{minipage}[t]{0.98\columnwidth}

\begin{center}
{\sc Operator Representation}
\end{center}
\vspace*{-1mm}

The input-output relation (\ref{eq:spreading_io}) describes a large class of linear operators.  This can easily be proved for a finite-dimensional setting using basic linear algebra.
We define the $N\times N$ cyclic time-shift matrix ${\bf D}$, which has ones in the \rev{subdiagonal} and in the top right corner and zero entries else,
and the diagonal $N\,\times\,N$ modulation (frequency shift) matrix ${\bf M}$, which has $e^{-j2\pi (i-1)/N}\!$, $i \in \{1,...,N\},$ as its $i$th main diagonal entry.
The inner product on $\C^{N\,\times\,N}$ is defined as
$
\langle{\bf A},{\bf B}\rangle=\mbox{Tr}\,({\bf B}^{H}\!{\bf A}),
$
where $\mbox{Tr}\,(\cdot)$ denotes the trace and the superscript $^H$ stands for Hermitian transposition.
It can be shown that the $N^2$ matrices $\big\{ \frac{1}{\sqrt{N}}{\bf M}^{l}{\bf D}^{m}\big\}_{m,l \in \{0,...,N-1\}}$
form a complete orthonormal set for $\C^{N\times N}$. Hence, every ${\bf H}\,\in\,\C^{N\times N}$ can be decomposed as
\begin{equation}\label{eq:sfdiscrete}
\Hm  = \sum_{m=0}^{N-1} \sum_{l=0}^{N-1} S_{\Hm}[m,l]\, \Mm^l\Dm^m
\end{equation}
with $S_{\Hm}[m,l] = \langle \Hm, {\bf M}^{l}{\bf D}^{m} \rangle/N$. This is the discrete, finite-dimensional counterpart of \eqref{eq:spreading_io}.

\end{minipage}
\end{center}
\vspace*{-1mm}
\end{floatbox}

\vspace{1mm}

\subsubsection{Channel Spread and Underspread Property}
Most channels are {underspread}, i.e., the amount of delay-Doppler spreading they induce is small in that their spreading function $S_{\H}(\tau,\nu)$
is effectively confined to a small region in the delay-Doppler plane. An example is visualized in the bottom center plot in Fig.~\ref{fig:intro}.
Selected aspects of the underspread property are considered in the box \textsc{Underspread Channels}, again in a finite-dimensional setting.

For spreading functions with finite support, a formal definition of the underspread property
can be obtained by circumscribing the support region with a rectangle that is centered around the
origin\footnote{The center of the rectangle is immaterial for the definition of the underspread property and is chosen to be the origin
for simplicity of exposition.}
and whose side lengths equal twice the channel's maximum delay $\tau_{\max}$ and maximum Doppler frequency $\nu_{\max}$, respectively.
The area of this rectangle, $d_\H = 4\tau_{\max}\nu_{\max}$, measures the channel's overall TF dispersion and is referred to as the channel spread.
A channel is then said to be underspread if $d_\H \le 1$ and overspread if $d_\H > 1$.  For spreading functions that do not have finite support,
the channel spread can be quantified in terms of moments \cite{GM-Hla-jmp98}.

For multipath propagation we have $d_\H \propto 1/c^2$ \cite{ltvbook11_ch1}. Hence, the channel spread of radio channels
(where $c$ equals the speed of light) is typically much smaller than that of underwater acoustic channels (where $c$ equals the speed of sound).
In fact, radio channels have $d_\H$ on the order of $10^{-6}$ to $10^{-3}$ and thus are highly underspread, whereas underwater acoustic channels
can even be overspread.

\begin{floatbox}[t!]
\vspace*{1mm}
\begin{center}
\begin{minipage}[t]{0.98\columnwidth}

\begin{center}
{\sc Underspread Channels}
\end{center}
\vspace*{-1mm}

Consider a finite-dimensional channel $\Hm \in \mathbb{C}^{N\times N}\!$, and assume that the discrete spreading function $S_\Hm[m,l]$
in \eqref{eq:sfdiscrete} is supported on a small set $\mathcal{S}$ around the origin, i.e., $S_\Hm[m,l]=0$ for $(m,l)\not\in\mathcal{S}$.
The sum in \eqref{eq:sfdiscrete} then consists of only $|\mathcal{S}|$ nonzero terms. The channel is underspread if $|\mathcal{S}|\le N$,
i.e., the number $|\mathcal{S}|$ of degrees of freedom of the channel does not exceed the dimensionality $N$ of the
ambient signal space.

A key observation explaining many properties of underspread channels is the fact that for small $m$ and small $l$,
time shifts $\Dm^{m}$ and frequency shifts $\Mm^{l}$ commute approximately. Specifically, using the (non-)commutation relation
$\Mm^{l}\Dm^{m} = \Dm^{m}\Mm^{l}e^{j2\pi\frac{ml}{N}}\hspace*{-.1mm}$ and the bound $|1\hspace*{-.1mm}-\hspace*{-.1mm} e^{j2\pi\phi}| \le 2\pi|\phi|$, one obtains
\begin{equation}\label{eq:TFcommute}
\|\Dm^{m}\Mm^{l} - \Mm^{l}\Dm^{m} \| \le 2\pi\frac{|ml|}{N}  \|\Dm^{m}\Mm^{l}\| ,
\end{equation}
where $\|\cdot\|$ is an arbitrary matrix norm. Clearly, if $|ml|$ is small relative to $N$, \eqref{eq:TFcommute} implies $\Dm^{m}\Mm^{l} \approx \Mm^{l}\Dm^{m}$.
\revv{In combination with \eqref{eq:sfdiscrete}, this approximate commutation property implies}
that underspread channels commute approximately.

We next demonstrate the approximate multiplicativity property \eqref{eq:appmult} in the finite-dimensional setting.
Here, the discrete TF transfer function $L_\Hm[n,k]$ equals the discrete two-dimensional Fourier transform of the discrete spreading function $S_\Hm[m,l]$,
i.e., $L_\Hm[n,k] = \sum_{m=0}^{N-1}\sum_{l=0}^{N-1} S_\Hm[m,l] e^{-j2\pi\frac{km-nl}{N}}$ (cf.~\eqref{eq:transspreadrel}).
For underspread channels $\Hm_1$ and $\Hm_2$ with identical spreading function support $\mathcal{S}$,
the approximation $L_{\Hm_1\Hm_2}[n,k]\approx L_{\Hm_1}[n,k]L_{\Hm_2}[n,k]$ (cf.\ \eqref{eq:appmult}) translates into the approximate convolution
\begin{equation}\label{eq:TFconv}
S_{\Hm_1\Hm_2}[m,l] \approx \! \sum_{(m',l')\in\mathcal{S}}
S_{\Hm_1}[m\hspace*{-.1mm}-\hspace*{-.1mm} m',l\hspace*{-.1mm}-\hspace*{-.1mm} l'] S_{\Hm_2}[m',l'] .
\end{equation}
To prove \eqref{eq:TFconv}, we start with the expression $S_{\Hm_1\Hm_2}[m,l] = \langle \Hm_1\Hm_2, {\bf M}^{l}{\bf D}^{m} \rangle/N$
and replace $\Hm_1$ and $\Hm_2$ by their spreading representations \eqref{eq:sfdiscrete}. This yields
\begin{multline}
\!\!\!S_{\Hm_1\Hm_2}[m,l] = \! \sum_{(m'',\,l'')\in\mathcal{S}} \! S_{\Hm_1}[m'',l''] \sum_{(m',\,l')\in\mathcal{S}} \! S_{\Hm_2}[m',l']\\
\cdot \langle \Mm^{l''}\Dm^{m''}\Mm^{l'}\Dm^{m'} \!, {\bf M}^{l}{\bf D}^{m} \rangle / N .
\label{eq:S_H_12}
\end{multline}
Now, thanks to \eqref{eq:TFcommute} and the orthogonality of the TF shift matrices ${\bf M}^{l}{\bf D}^{m}$, we obtain
$\langle \Mm^{l''}\Dm^{m''}\Mm^{l'}\Dm^{m'}, {\bf M}^{l}{\bf D}^{m} \rangle/N \!\approx\! \delta_{(l-l'-l'') \,{\rm mod}\, N} \, \delta_{(m-m'-m'') \,{\rm mod}\, N}$.
Inserting this into \eqref{eq:S_H_12} yields \eqref{eq:TFconv}.

\end{minipage}
\end{center}
\vspace{-1mm}
\end{floatbox}

\vspace{1.5mm}

\subsubsection{TF Transfer Function}
The spreading function represents channels in the delay-Doppler domain. A dual TF representation, termed {TF transfer function}, represents
channels in the TF domain and is defined as the two-dimensional (2-D) Fourier transform of the spreading function \cite{Zad50,ltvbook11_ch1,GM-Hla-jmp98}:
\begin{equation}\label{eq:transspreadrel}
L_\H(t,f) = \iint S_\H(\tau,\nu) \, e^{-j2\pi(f\tau-t\nu)}\,d\tau\,d\nu.
\end{equation}
For underspread channels, $L_\H(t,f)$ is smooth and characterizes the channel's TF weighting behavior.
This generalizes the frequency response $H(f)$ of time-invariant channels.

The complex exponentials $x(t) = e^{j2\pi f_0 t}$, $f_0 \!\in\! \mathbb{R}$, are eigen\-functions of all linear time-invariant channels.
For TF dispersive channels, a universal set of structured eigenfunctions does not exist. Underspread channels, however,
satisfy the approximate eigenrelation
\begin{equation}\label{eq:appeigen}
\big(\H g_{t_0,f_0}\big)(t) \approx L_\H(t_0,f_0) \, g_{t_0,f_0}(t) ,
\end{equation}
with $g_{t_0,f_0}(t) = \big( \M_{f_0}\D_{t_0}g \big)(t)$; the accuracy of \eqref{eq:appeigen} depends on how well the function $g(t)$ is localized
(around time $0$ and frequency $0$). Thus, $L_\H(t_0,f_0)$ is the approximate eigenvalue associated with an approximate eigenfunction that is
TF localized around the TF point $(t_0,f_0)$.
This property entails an approximate diagonalization of the channel and explains why OFDM is a natural choice for signaling over
underspread TF dispersive channels.

For underspread channels, the TF transfer function is furthermore approximately multiplicative, i.e.,
\begin{equation}\label{eq:appmult}
L_{\H_1\H_2}(t,f) \approx L_{\H_1}(t,f) L_{\H_2}(t,f) .
\end{equation}
This implies that underspread channels commute approximately, i.e., $\H_1\H_2\approx\H_2\H_1$.
The approximate commutation of underspread channels  is of practical importance, e.g., in channel sounding \cite{sounder_tc00}.
A derivation of \eqref{eq:appmult} in the finite-dimensional setting is given in the box \textsc{Underspread Channels}.

The approximations \eqref{eq:appeigen} and \eqref{eq:appmult} nicely show that, in terms of transfer function calculus,
underspread TF dispersive channels behave approximately like time-invariant channels. This is due to the fact that underspread channels
share a structured set of approximate eigenfunctions.

In Fig.~\ref{fig:channel_reps}, we show the TF transfer function and spreading function of a realization of Channel 6 specified in the DRM standard \cite{etsi_drm}.
This is an underspread channel that models skywave propagation. The echoes visible in Fig.~\ref{fig:channel_reps}(b)
correspond to multiple reflections at the ionosphere.

\vspace{-1mm}

\subsection{Channel Statistics}

Many channels are modeled as random; examples of the underlying phenomena include fading, unknown time and frequency offsets, and phase noise.
The system functions  $S_\H(\tau,\nu)$ and $L_{\H}(t,f)$ then become 2-D random processes, with 4-D correlation (or covariance) functions.

\vspace{1.5mm}

\subsubsection{WSSUS Channels and Scattering Function}

An important simplification of the channel statistics is obtained for channels that are wide-sense stationary with uncorrelated scattering (WSSUS). Here,
$L_\H(t,f)$ is a process that is stationary in time and frequency. Hence, its correlation function is independent of $t$ and $f$, i.e.,
\[
\mathsf{E}\big\{ L_\H(t,f)\,L_\H^*(t-\Delta t,f-\Delta f)  \big\} = R_\H(\Delta t, \Delta f),
\]
where $R_\H(\Delta t, \Delta f)$ is known as the channel's TF correlation function. Correspondingly, the scatterer reflectivities described by the
spreading function are uncorrelated, i.e.,
\[
\mathsf{E}\big\{ S_\H(\tau,\nu)\,S_\H^*(\tau',\nu')  \big\} = C_\H(\tau,\nu)\,\delta(\tau-\tau')\,\delta(\nu-\nu').
\]
Here, $C_\H(\tau,\nu)\ge 0$ describes the average intensity of scatterers with delay $\tau$ and Doppler shift $\nu$ and is referred to as
scattering function \cite{Bel63,ltvbook11_ch1}. The scattering function and the TF correlation function are related via a 2-D Fourier transform:
\[
C_\H(\tau,\nu) = \iint R_\H(\Delta t, \Delta f)\,e^{-j2\pi(\nu\Delta t -\tau\Delta f)}\,d\Delta t\,d\Delta f.
\]
This shows that $C_\H(\tau,\nu)$ can be interpreted as the delay-Doppler domain power spectral density of $L_\H(t,f)$.

\begin{figure}[t]
\renewcommand\baselinestretch{1.1}\small\normalsize
\begin{center}
\noindent\includegraphics[width=\columnwidth]{channel_reps.pdf}
\end{center}
\vspace*{-5mm}
\caption{\label{fig:channel_reps}
Example of an underspread TF dispersive channel with maximum delay $6$\,ms and Doppler spread $14.4$\,Hz:
(a) TF transfer function over a duration of $1$\,s and a bandwidth of $2.5$\,kHz,
(b) spreading function in the delay-Doppler region $[0,8)$\,ms$\times[-40,40)$\,Hz
(outside this region, the spreading function is 40\,dB below the maximum value).
For both representations, the magnitude is displayed on a log scale with 40\,dB dynamic range.}
\vspace*{-2mm}
\end{figure}

The scattering function relates the time-varying power spectra of the transmit and receive signals according to \cite{ltvbook11_ch1}
\[
P_y(t,f) = \iint C_\H(\tau,\nu)\,P_{\M_\nu\D_\tau x}(t,f)\,d\tau\,d\nu,
\]
where $P_\cdot(t,f)$ is an arbitrary type I TF energy spectrum \cite{GM_spectra02}. This relation is the statistical TF counterpart of \eqref{eq:spreading_io};
it amounts to a convolution, since $P_{\M_\nu\D_\tau x}(t,f) = P_{x}(t-\tau,f-\nu)$.

A WSSUS random channel is said to be underspread if its scattering function $C_\H(\tau,\nu)$ is effectively confined to a small delay-Doppler region
(the spreading function of every channel realization then is confined to the same region). Wireless (radio) channels are underspread also in this stochastic sense.

\vspace{1.5mm}

\subsubsection{Non-WSSUS Channels and Local Scattering Function}
Recently, high-mobility applications like vehicular communications have spurred interest in non-WSSUS channels \cite{ltvbook11_ch1,cfm_proc11}.
For non-WSSUS channels, $L_\H(t,f)$ is a nonstationary random process and different scatterer contributions are correlated.
A generalization of the scattering function $C_\H(\tau,\nu)$ to non-WSSUS channels is provided by the local scattering function \cite{ltvbook11_ch1},
which equals the (generalized) Wigner-Ville spectrum \cite{fla-book2,GM_spectra02} of $L_\H(t,f)$.
The local scattering function $\mathcal{C}_\H(t,f;\tau,\nu)$ describes the average power of scatterers that cause a delay $\tau$ and a Doppler shift $\nu$
of the transmit signal component localized around time $t$ and frequency $f$.

\vspace*{-1mm}

\section{OFDM}

\vspace*{.5mm}

In the spirit of Gabor \cite{Gabor46a}, OFDM transmits data symbols via TF logons (TF shifts of a prototype pulse).
OFDM is used in a large number of wireless and wireline communication systems and standards.
Among other reasons, OFDM is popular because cyclic prefix (CP) OFDM diagonalizes time-invariant channels
and, more generally, well TF localized WH sets approximately diagonalize underspread TF dispersive channels.
Here, we consider \emph{pulse-shaping} OFDM, which constitutes a unified framework for CP-OFDM \cite{Peled80}, zero-padded OFDM \cite{wanggg00},
DFT filterbank modulation \cite{FBMC_JASP10}, and, with a slight modification, OFDM with offset quadrature amplitude modulation (OQAM) \cite{boel_spie99}.

\vspace*{-1mm}

\subsection{Modulation and Demodulation}

The transmit signal in a pulse-shaping OFDM system is formed by modulating data symbols $c_{n,k}$ onto TF shifted versions of a transmit pulse $g(t)$
(e.g., \cite{kozmol98,hanzo-mimo-ofdm,Matz-Charlypaper07,boel_spie99}):
\begin{equation}\label{eq:ofdmmod}
x(t) = \sum_{n\in\Z} \sum_{k\in\Z} c_{n,k}\,g_{n,k}(t),
\vspace*{-2mm}
\end{equation}
with
\vspace*{-1mm}
\[
g_{n,k}(t) = \big( \M_{kF}\D_{nT} g \big)(t) = g(t-nT)e^{j2\pi kFt} .
\vspace*{.5mm}
\]
Here, $T$ is the OFDM symbol duration and $F$ is the subcarrier spacing. (In practical OFDM systems, the sum with respect to $k$ involves only a finite number of subcarriers.
We assume infinitely many subcarriers to simplify the presentation.) The collection of ``logons'' $\{g_{n,k}(t)\}_{n,k \in \mathbb{Z}}$ is known as a WH set.
Its TF localization structure is schematically illustrated in the bottom left plot in Fig.~\ref{fig:intro} (in reality, the $g_{n,k}(t)$ overlap in time or in frequency).
To recover the data symbols $c_{n,k}$, the receiver projects the receive signal $y(t)$ onto TF shifted versions of a receive pulse $\gamma(t)$
by computing the inner products
\begin{equation}\label{eq:ofdmdemod}
\hat{c}_{n,k}=\big<y,\gamma_{n,k}\big>=\int y(t)\,\gamma_{n,k}^*(t)\,dt,
\end{equation}
with $\gamma_{n,k}(t) = \big( \M_{kF}\D_{nT} \gamma \big)(t) = \gamma(t-nT)e^{j2\pi kFt}$. This OFDM demodulation is followed by further receiver processing such as
channel estimation, demapping, and decoding.

In the absence of channel distortions and noise,
\revv{it is desirable to have}
perfect symbol recovery,
i.e., $\hat{c}_{n,k} \!=\! {c}_{n,k}$\revv{; this} is guaranteed if the WH sets
$\{g_{n,k}(t)\}_{n,k \in \mathbb{Z}}$ and $\{\gamma_{n,k}(t)\}_{n,k \in \mathbb{Z}}$ satisfy the biorthogonality condition
$\langle g_{n,k},\gamma_{n^{\prime}\!,k^{\prime}} \rangle = \delta_{n-n^{\prime}}\delta_{k-k^{\prime}}$. Biorthogonality presupposes $TF \!>\! 1$,
in which case the system is said to employ a TF guard region \cite{Haas96}. CP-OFDM \cite{Peled80} and zero-padded OFDM \cite{wanggg00}
are special cases, with the TF guard region being a temporal guard region only. However, $TF\!>\! 1$ can also be achieved by introducing a spectral guard region via an
increase of the subcarrier spacing $F$; this can reduce intercarrier interference in frequency-dispersive environments.
The spectral efficiency of an OFDM system is inversely proportional to $TF$ and is thus determined by the density of the TF grid $\big\{(nT,kF)\big\}_{n,k\in\Z}$.

\vspace*{-1mm}

\subsection{Analysis-Synthesis Duality and WH Frames}

WH frames are complete or overcomplete (i.e., redundant) WH sets with a certain guaranteed numerical stability of reconstruction
\cite{groebook,Fei_Str_97,christensen03} (see the box \textsc{Weyl-Heisenberg Frames}). When decomposing a signal $x(t)$ into a WH frame
$\{g_{n,k}(t)\}$ with dual WH frame $\{\gamma_{n,k}(t)\}$, we would first compute the expansion coefficients $\langle x,\gamma_{n,k} \rangle$
(analysis stage) and then reconstruct $x(t)$ according to $x(t)=\sum_{n\in\Z}\sum_{k\in\Z} \hspace*{.1mm} \langle x,\gamma_{n,k} \rangle g_{n,k}(t)$ (synthesis stage).
In OFDM systems, the transmitter performs \emph{synthesis} of the transmit signal according to
\revv{\eqref{eq:ofdmmod}}
with the data symbols $c_{n,k}$ playing the role of the expansion coefficients, and the receiver performs \emph{analysis} according to
\revv{\eqref{eq:ofdmdemod}}.
This apparent duality is closely related to the duality and biorthogonality theory for WH frames \cite{wex90,janjf95,daubjf95}.

\begin{floatbox}[t!]
\vspace*{1mm}
\begin{center}
\begin{minipage}[t]{0.98\columnwidth}

\begin{center}
{\sc Weyl-Heisenberg Frames}
\end{center}
\vspace*{-1mm}

For $g(t) \!\in\! L_{2}(\R)$ and $T,F\!>\!0$, a function set $(g,T,F) = \{ g(t-nT)e^{j2\pi kFt} \}_{n,k\in\Z}$ is called a WH frame or Gabor frame for $L_2(\R)$
if for all $x(t)\in L_2(\R)$
\[
A\|x\|^{2}\,\le\,\sum_{n\in\Z} \sum_{k\in\Z} |\langle x,g_{n,k} \rangle|^{2}\,\le\,B\|x\|^{2}
\]
with $0 \!<\! A \!\le\! B \!<\!  \infty$ \cite{groebook,Fei_Str_97,christensen03}. In what follows, we use the tightest constants $A$ and $B$; these are called
lower and upper frame bound, respectively. The frame operator $\S$ is defined as the positive definite linear operator that maps $L_{2}(\R)$ onto $L_{2}(\R)$
according to
\[
({\S}x)(t)= \sum_{n\in\Z} \sum_{k\in\Z} \langle x,g_{n,k} \rangle g_{n,k}(t).
\]
For a WH frame $(g,T,F)$, the (minimal) {dual WH frame} is given by the set $(\gamma,T,F)$, where $\gamma(t)=({\S}^{-1}g)(t)$.
The lower and upper frame bounds of the dual frame are given by $1/B$ and $1/A$, respectively. Using dual WH frames $(g,T,F)$ and $(\gamma,T,F)$,
every signal $x(t) \in L_{2}(\R)$ can be decomposed as
\begin{equation}
x(t) = \sum_{n\in\Z} \sum_{k\in\Z} \langle x,\gamma_{n,k} \rangle g_{n,k}(t)
= \sum_{n\in\Z} \sum_{k\in\Z} \langle x,g_{n,k} \rangle \gamma_{n,k}(t) .
\label{eq:frame_decomp}
\end{equation}

A WH frame is called tight if $A \!=\! B$. For a tight WH frame, we have ${\S}=A{\mathds I}$, where ${\mathds I}$ is the identity operator,
and hence $\gamma(t)=\frac{1}{A}g(t)$. If $(g,T,F)$ is a WH frame, $({\S}^{-1/2}g,T,F)$ is a tight WH frame with $A \!=\! B \!=\! 1$.
Here, ${\S}^{-1/2}$ is the inverse positive definite square root of ${\S}$.

In general, it is difficult to determine whether a given WH set $(g,T,F)$ is a WH frame. Intuitively, choosing $T$ and $F$ too large leaves ``gaps'' in $L_{2}(\R)$.
Indeed, it can be shown that for $g(t) \in L_{2}(\R)$ and $TF>1$, the WH set $(g,T,F)$ cannot be a frame for $L_2(\R)$.
The elements $g_{n,k}(t)$ of a WH frame with $TF=1$ are necessarily linearly independent, whereas WH frames with $TF<1$ necessarily
have linearly dependent elements $g_{n,k}(t)$.
Therefore, $(g,T,F)$ can be a frame for $L_2(\R)$ only if $TF \le 1$, i.e., when the TF grid
$\big\{(nT,kF)\big\}_{n,k\in\Z}$ is sufficiently dense. We note that WH analysis and synthesis can be interpreted as
the analysis and synthesis stage, respectively, of a DFT filter bank \cite{vaid93}.

\end{minipage}
\end{center}
\vspace*{-1mm}
\end{floatbox}

Duality and biorthogonality theory states that the WH sets $(g,T,F) \!=\! \{g(t \!-\! nT)e^{j2\pi kFt}\}_{n,k\in\Z}$ and $(\gamma,T,F)$
are biorthogonal if and only if the associated WH sets $(g,1/F,1/T)$ and $(\gamma,1/F,1/T)$ are dual frames; furthermore,
the WH set $(g,T,F)$ is orthogonal if and only if the associated WH set $(g,1/F,1/T)$ is a tight frame
(cf.\ the box \textsc{Duality and Biorthogonality}). The design of biorthogonal and orthogonal OFDM systems is therefore reduced to the
widely studied problem of designing, respectively, dual and tight WH frames \cite{daub92}.

\begin{floatbox}[t!]
\vspace*{1mm}
\begin{center}
\begin{minipage}[t]{0.98\columnwidth}

\begin{center}
{\sc Duality and Biorthogonality}
\end{center}
\vspace*{-1mm}
In the finite-dimensional (cyclic) case, the duality and biorthogonality relation for WH frames essentially follows from the Poisson summation formula \cite{wex90}.
We take all signals to be discrete-time and $N$-periodic and consider the WH frames $\{g_{n,k}[\ii]=g[\ii-nL]e^{j 2\pi\frac{k\ii}{M}}\}_{n\in \{0,\ldots,N/L-1\}, \, k\in \{0,\ldots,M-1\}}$
and $\{\gamma_{n,k}[\ii]=\gamma[\ii-nL]e^{j 2\pi \frac{k\ii}{M}} \}_{n\in \{0,\ldots,N/L-1\}, \, k\in \{0,\ldots,M-1\}}$
with time-shift parameter $L$ and frequency-shift parameter $1/M$, where $L,M\in \mathbb{N}$ and $M \ge L$. We assume that $N$ is an integer multiple of both $L$ and $M$.

We want to show that the WH frames $\{g_{n,k}[\ii]\}$ and $\{\gamma_{n,k}[\ii]\}$ are dual if and only if the WH sets
$\{\tilde{g}_{n,k}[\ii]=g[\ii-nM]e^{j 2\pi \frac{k\ii}{L}} \}_{n\in \{0,\ldots,N/M-1\}, \, k\in \{0,\ldots,L-1\}}$ and
$\{\tilde{\gamma}_{n,k}[\ii]=\gamma[\ii-nM]e^{j 2\pi \frac{k\ii}{L}} \}_{n\in \{0,\ldots,N/M-1\}, \, k\in \{0,\ldots,L-1\}}$
with time-shift parameter $M$ and frequency-shift parameter $1/L$ are biorthogonal, i.e.,
$$
\langle \tilde{g}_{n,k},\tilde{\gamma}_{n^\prime,k^\prime}\rangle=\frac{L}{M} \, \delta_{n-n^\prime} \hspace*{.1mm} \delta_{k-k^\prime} \,.
$$

We start by noting that duality of $\{g_{n,k}[\ii]\}$ and $\{\gamma_{n,k}[\ii]\}$ (cf.\ \eqref{eq:frame_decomp}) is equivalent to the completeness relation
\begin{equation}
\sum_{n=0}^{N/L-1}\sum_{k=0}^{M-1} g_{n,k}[\ii]  \gamma^{\ast}_{n,k}[\ii^\prime] = \delta_{\ii-\ii^{\prime}} \, .
\label{eq:complete}
\end{equation}
The left-hand side of this relation can be shown to equal
\begin{equation}
M \! \sum_{n=-\infty}^{\infty} \!\Bigg[ \delta_{\ii-\ii^{\prime}-nM}
\! \sum_{n'=0}^{N/L-1} \! f_{n}[\ii-n'L] \Bigg] \hspace*{.1mm} ,
\label{eq:lhs}
\end{equation}
where $f_n[\ii] = g[\ii]\gamma^{\ast}[\ii-nM]$. Furthermore, the Poisson summation formula yields
\begin{equation}
\sum_{n'=0}^{N/L-1} \! f_n[\ii-n'L]
= \frac{1}{L}\sum_{k=0}^{L-1} F_n\bigg[\frac{kN}{L}\bigg]\,e^{j2\pi\frac{\ii k}{L}} ,
\label{eq:poisson}
\end{equation}
with $F_{n}[k]\,= \sum_{\ii=0}^{N-1} f_n[\ii]e^{-j2\pi\frac{k\ii}{N}}$.
Realizing that $F_n[kN/L]$\linebreak
$= \langle\tilde{g},\tilde{\gamma}_{n,k}\rangle$,
and inserting into \eqref{eq:poisson} and, in turn, \eqref{eq:lhs}, we see that the left-hand side of \eqref{eq:complete} equals
\[
\frac{M}{L} \! \sum_{n=-\infty}^{\infty} \!\Bigg[ \delta_{\ii-\ii^{\prime}-nM} \sum_{k=0}^{L-1} \langle\tilde{g},\tilde{\gamma}_{n,k}\rangle \,e^{j2\pi\frac{\ii k}{L}} \Bigg] \hspace*{.1mm} .
\]
Thus, we can conclude that $\{g_{n,k}[i]\}$ and $\{\gamma_{n,k}[i]\}$ are dual if and only if $\langle \tilde{g},\tilde{\gamma}_{n,k}\rangle=\frac{L}{M} \, \delta_{n} \delta_{k}$,
i.e., if and only if the WH sets $\{\tilde{g}_{n,k}[i]\}$ and $\{\tilde{\gamma}_{n,k}[i]\}$ are biorthogonal.
\end{minipage}
\end{center}
\vspace*{-1mm}
\end{floatbox}

\vspace*{-1mm}

\subsection{Effect of a Doubly Dispersive Channel}

Consider an OFDM system with transmit pulse $g(t)$ and receive pulse $\gamma(t)$.
The transmit signal $x(t)$ is distorted by a TF dispersive channel $\H$ and contaminated by additive noise $w(t)$,
resulting in the receive signal $y(t)=({\H}x)(t)+w(t)$. The OFDM demodulator output $\hat{c}_{n,k} =\langle y,\gamma_{n,k}\rangle$ then equals
\begin{equation}\label{eq:isiici_0}
\hat{c}_{n,k} \hspace*{.1mm}\ist=\, {H}_{n,k}\,c_{n,k} + I_{n,k} + w_{n,k} \,,
\end{equation}
where $H_{n,k}=\langle {\H}g_{n,k}, \gamma_{n,k} \rangle$ is the complex gain factor affecting the desired symbol $c_{n,k}$,
$I_{n,k}$ summarizes the interference caused by all other symbols $c_{n^{\prime}\!,k^{\prime}}$, $(n^{\prime}\!,k^{\prime}) \neq (n,k)$, and
$w_{n,k}=\langle w,\gamma_{n,k} \rangle$. The interference term $I_{n,k}$ is given by
\begin{equation}\label{eq:isiici}
I_{n,k} \hspace*{.1mm}=\!
\sum_{(n'\!,k')\,\in\, \Z^2\backslash (n,k)} \!\!\langle {\H}g_{n'\!,k'}, \gamma_{n,k} \rangle \,c_{n^{\prime}\!,k^{\prime}} \,.
\end{equation}
This comprises interference from symbols at different times $n'\hspace*{-.1mm}\neq n$ (intersymbol interference, ISI) and at different frequencies $k'\hspace*{-.1mm}\neq k$
(intercarrier interference, ICI).

ISI and ICI are negligible if $\H$ is underspread and $g(t)$, $\gamma(t)$, $T$, and $F$ are chosen appropriately as discussed in the next subsection.
In that case, the input-output relation \eqref{eq:isiici_0} decouples into a set of non-interfering parallel scalar channels according to
\begin{equation} \label{eq:diag}
\hat{c}_{n,k} \approx H_{n,k} \,c_{n,k} + w_{n,k} \,.
\end{equation}
This {approximate diagonalization of an underspread channel $\H$ drastically simplifies receiver tasks like data detection and channel estimation.
Note that $g_{n,k}(t)$ and $\gamma_{n,k}(t)$ can be viewed as {approximate singular functions} of $\H$ and $H_{n,k}=\langle {\H}g_{n,k}, \gamma_{n,k} \rangle$
as the corresponding approximate singular values. For normalized pulses $g(t)$ and $\gamma(t)$, it can be shown that $H_{n,k} \approx L_\H(nT,kF)$.
In the case of a time-invariant channel, CP-OFDM turns linear convolution into cyclic convolution. The corresponding channel matrix is
circulant and diagonalized by the FFT (on which CP-OFDM is based) \cite{Peled80}, so that \eqref{eq:diag} becomes exact.

\vspace*{-1mm}

\subsection{Pulse Design}

Next, we consider the problem of designing the transmit pulse $g(t)$ and the receive pulse $\gamma(t)$ such that
small ISI and ICI are obtained \cite{boel_spie99,Matz-Charlypaper07,schniter_tsp07}. We note that OFDM systems with sophisticated ISI/ICI-reducing pulse shapes
are currently hardly used in practice. This can be attributed to the fact that ISI and ICI can alternatively be mitigated
using equalization \cite{schniter_tsp07}.

For WSSUS channels, the mean power of the interference term $I_{n,k}$ in \eqref{eq:isiici} can be shown to equal \cite{Matz-Charlypaper07}
\[
P_I = \mathsf{E}\{ |I_{n,k}|^2 \} = \iint C_\H(\tau,\nu)\,\tilde{A}_{g,\gamma}(\tau,\nu)\,d\tau\,d\nu ,
\vspace*{-1mm}
\]
with
\[
\tilde{A}_{g,\gamma}(\tau,\nu) \hspace*{.1mm}=\!  \sum_{ (m,l)\,\in\, \Z^2\backslash(0,0)} \!\! |A_{g,\gamma}(\tau-mT,\nu-lF)|^2,
\]
where $A_{g,\gamma}(\tau,\nu) =\hspace*{-.1mm} \int g(t)\,\gamma^*(t \!-\! \tau)\,e^{-j2\pi\nu t}\hspace*{.1mm} dt
=\langle g,\M_\nu\D_\tau \gamma\rangle$ is the cross-ambiguity function of $g(t)$ and $\gamma(t)$ \cite{fla-book2,groebook}.
To obtain small ISI/ICI power $P_I$, the translates $|A_{g,\gamma}(\tau-mT,\nu-lF)|^2$, $(m,l) \in\Z\backslash (0,0)$,
should have little overlap with the channel's scattering function $C_\H(\tau,\nu)$.
Clearly, making $P_I$ small by suitably choosing $g,\gamma,T$, and $F$ is easier for underspread channels with $C_\H(\tau,\nu)$
better concentrated around $(0,0)$. We then have to design
pulses $g(t)$ and $\gamma(t)$ for which
$A_{g,\gamma}(\tau,\nu)$ decays rapidly, which in turn requires that the pulses be well TF localized \cite{Matz-Charlypaper07}.
An example of well-localized biorthogonal pulses is shown in Fig.~\ref{fig:pulses_ambifunc}.
Biorthogonality implies $A_{g,\gamma}(mT,lF) \!=\! \delta_m \delta_l$ and indeed
the zeros of $A_{g,\gamma}(\tau,\nu)$ for $(\tau,\nu)=(mT,lF)$, $(m,l) \in\Z\backslash (0,0)$,
are clearly visible in Fig.~\ref{fig:pulses_ambifunc}(c).
Further numerical results for pulse designs and the associated ISI/ICI levels are provided in \cite{Matz-Charlypaper07}.

\begin{figure}
\renewcommand\baselinestretch{1.1}\small\normalsize
\begin{center}
\noindent\includegraphics[width=\columnwidth]{pulses_ambifunc2.pdf}
\end{center}
\vspace*{-5mm}
\caption{\label{fig:pulses_ambifunc}
Example of a biorthogonal pulse-shaping OFDM system with $TF=1.33$, optimized for a TF dispersive channel with $\tau_{\max}=T/10$ and $\nu_{\max}=F/24$:
(a) transmit pulse $g(t)$, (b) receive pulse $\gamma(t)$, and (c) magnitude of the cross-ambiguity function $A_{g,\gamma}(\tau,\nu)$ (displayed on a log scale).}
\vspace*{-1mm}
\end{figure}

The analysis above shows that small ISI/ICI power $P_I$ requires an underspread channel $\H$ and pulses $g(t)$ and $\gamma(t)$
that are well localized both in time and in frequency. CP-OFDM employs a rectangular $g(t)$ (usually with a slight roll-off), whose
excellent time localization is well suited to purely time-dispersive channels; however, its poor frequency localization leads to ICI in frequency-dispersive channels.
Since well TF localized WH frames $(g,T,F)$ exist only for $TF \!<\! 1$ \cite{groebook}, it follows from duality and biorthogonality theory that well TF localized
biorthogonal WH sets $(g,T,F)$ and $(\gamma,T,F)$ (which result in low ISI/ICI) exist only for $TF \!>\! 1$. Thus, there is a trade\-off between spectral efficiency and
the amount of ISI/ICI incurred. Specifically, if $(g,T,F)$ with $TF \!=\! 1$ (maximal spectral efficiency) is an orthogonal basis for $L_{2}(\R)$,
then $g(t)$ and its Fourier transform $G(f)$ cannot satisfy both $\int t^{2}|g(t)|^{2}\,dt \hspace*{-.1mm}<\hspace*{-.1mm} \infty$ and $\int f^{2}|G(f)|^{2}\,df \hspace*{-.1mm}<\hspace*{-.1mm} \infty$
\cite{groebook}.

\vspace{-1mm}

\section{Channel Capacity}

We have seen that WH signal sets---corresponding to OFDM modulation---are well suited to communication over underspread TF dispersive channels since
they approximately diagonalize the channel. In addition, WH sets are also useful for characterizing the capacity limits of continuous-time WSSUS fading channels.
We consider the noncoherent setting where neither transmitter nor receiver knows the channel realization but the transmitter knows the
channel statistics (i.e., the scattering function $C_{\H}(\tau,\nu)$). The noncoherent capacity of fading channels \cite{gallager68,Ken69,bigshlom98}
is the ultimate limit on the achievable rate since overhead transmissions like pilots and training sequences reduce spectral efficiency.

The standard approach to information-theoretic analyses of continuous-time channels is to discretize the input-output relation through a projection
onto the singular functions of the channel \cite{gallager68}. This yields a diagonalized discretized channel with noninteracting scalar input-output relations,
similar to (\ref{eq:diag}). In the noncoherent case, this approach works only if all channel realizations have the same singular functions.
This is the case for time-invariant channels, where the eigenfunctions are complex sinusoids independently of the channel realization.
However, for TF dispersive channels, the singular functions generally depend on the channel realization and do not have a specific structure.

Nevertheless, approximate capacity expressions can be obtained by using the channel discretization induced by OFDM
\cite{durisiboel_book11}. We consider an underspread Gaussian WSSUS channel $\H$ with additive white Gaussian noise and
OFDM modulation and demodulation using an orthonormal WH set $(g \!=\! \gamma,T,F)$, where $TF > 1$ and $g$ is well TF localized.
An important advantage of using WH sets to discretize the channel (even though they do not diagonalize the channel exactly) is the fact that
the channel coefficients $H_{n,k}$ in \eqref{eq:isiici_0} inherit the 2-D stationarity property of the continuous-time WSSUS channel.
In the low signal-to-noise ratio (SNR) regime, ignoring the ISI/ICI term $I_{n,k}$ (cf.\ \eqref{eq:isiici_0}) in the capacity computation leads to small
approximation errors \cite{durisiboel_book11}. In the high-SNR regime, ISI/ICI cannot be neglected \cite{durisiboel_book11}.

Using OFDM-based channel discretization, the capacity for low SNR satisfies \cite{durisiboel_book11}
$$
\mathsf{C}(\rho) \,\approx\, \mathsf{C}_{\mbox{\footnotesize AWGN}}(\rho) \hspace*{.1mm}-\hspace*{-.1mm} \iint \log\big(1+\rho\,C_{\H}(\tau,\nu)\big)\, d\tau\hspace*{.1mm} d\nu,
$$
where $\mathsf{C}_{\mbox{\footnotesize AWGN}}(\rho)$ is the capacity of a nondispersive additive white Gaussian noise channel
and the SNR $\rho$ is inversely proportional to the bandwidth. We see that $\mathsf{C}(\rho)$ is approximately equal to $\mathsf{C}_{\mbox{\footnotesize AWGN}}(\rho)$
minus a penalty term that is due to the unknown channel and increases with increasing channel spread (i.e., effective support of $C_{\H}(\tau,\nu)$). Furthermore,
$\mathsf{C}(\rho) \!\to\! 0$ as the bandwidth grows large. Intuitively, because of the uncorrelated scattering nature of the channel, the number of independent diversity
branches increases as the channel spread or the signal bandwidth increases and thus the receiver can no longer resolve the corresponding channel uncertainty.
This also implies that $\mathsf{C}(\rho)$ has a maximum at a certain finite bandwidth. A detailed discussion of this phenomenon is provided in \cite{durisiboel_book11}.

For high SNR, $\mathsf{C}(\rho)$ is close to $\mathsf{C}_{\mbox{\footnotesize AWGN}}(\rho)$ for channel spreads occurring in wireless (radio) communications.
Information-theoretic guidelines for the design of $(g,T,F)$ reveal that choosing $TF$ slightly larger than $1$ and using a root-raised-cosine pulse
for $g$ yields a lower bound on $\mathsf{C}(\rho)$ that is very close to the upper bound given by $\mathsf{C}_{\mbox{\footnotesize AWGN}}(\rho)$ \cite{durisiboel_book11}.

\section{System Identification}

The goal of channel/system identification \cite{Kai62} is to determine a channel/system $\H$ from the output signal $y(t)=(\H x)(t)$
given knowledge of the sounding (or probing) signal $x(t)$. This is relevant to dedicated channel sounding/measurement \cite{sounder_tc00},
channel estimation in the course of data transmission, and numerous other applications such as radar and sonar \cite{strohmer09}.
Let us consider a TF dispersive channel $\H$ with spreading function $S_{\H}(\tau,\nu)$ supported in $[-\tau_{\max},\tau_{\max})\times [-\nu_{\max},\nu_{\max})$.
In a practical scenario with finite input signal bandwidth $B$ and finite output signal observation time $D$, the input-output relation \eqref{eq:spreading_io}
is discretized, resulting in an input-output relation of the form $\vect{y} \!=\! \rev{\mathbf{X}}\mathbf{s}$ as explained in the box \textsc{Discretization}.

\begin{floatbox}[t!]
\vspace*{1mm}
\begin{center}
\begin{minipage}[t]{0.98\columnwidth}

\begin{center}
{\sc Discretization}
\end{center}
\vspace*{-1mm}

We consider a transmit signal $x(t)$ that is band-limited to $[-B/2,B/2)$, and we observe the receive signal $y(t) = ({\H}x)(t)$ on $[-D/2,D/2)$.
Then, for $t\in[-D/2,D/2)$, the input-output relation \eqref{eq:spreading_io} becomes \cite{Bel63}
\begin{equation}
y(t) \approx \frac{1}{BD} \sum_{m\in \Z}\sum_{l \in \Z} S_\H\! \left(\frac{m}{B},\frac{l}{D} \right) \!\big(\M_{l/D}\D_{m/B} x)(t) .
\label{eq:sfunc_discrete}
\end{equation}
Thus, band-limiting the input and time-limiting the output leads to a discretization of \eqref{eq:spreading_io} with
sample spacing $1/B$ and $1/D$ in delay and Doppler, respectively.
For random (i.e., fading) channels, based on \eqref{eq:sfunc_discrete}, the concept of TF coherence regions and a TF rake receiver are developed in \cite{sayeed99}.

If the spreading function $S_{\H}(\tau,\nu)$ is supported in $[-\tau_{\max},\tau_{\max})\times [-\nu_{\max},\nu_{\max})$,
only $|\mathcal{S}|=\lceil 4\tau_{\max}\nu_{\max}BD \rceil$ terms in \eqref{eq:sfunc_discrete} are non-zero. For $\nu_{\max} \hspace*{-.1mm}\rmv\ll\hspace*{-.1mm}\rmv B$,
the output signal $y(t)$ in \eqref{eq:sfunc_discrete} is approximately band-limited to $[-B/2,B/2)$. According to \cite{slepian76}, $y(t)$ restricted to $[-D/2,D/2)$
then lives in a signal space of dimension $N \!=\! \lceil BD \rceil$ that is spanned by an orthonormal basis of prolate spheroidal wave functions \cite{slepian76}.
Arranging the basis expansion coefficients of $y(t)$ in a vector $\vect{y}\in\C^{N}$, the input-output relation \eqref{eq:sfunc_discrete} translates into
\begin{equation}
\vect{y} = \rev{\mathbf{X}} \mathbf{s}.
\label{eq:syseqyAs}
\end{equation}
Here, $\mathbf{s}\in \C^{|\mathcal{S}|}$ contains the $|\mathcal{S}|$ samples $S_{\H}(m/B,l/D)$,
$(m/B,l/D) \in [-\tau_{\max},\tau_{\max})\times [-\nu_{\max},\nu_{\max})$,
and each column of $\rev{\mathbf{X}}\in \C^{N \times |\mathcal{S}|}$ contains the expansion coefficients of a TF shifted version $\big(\M_{l/D}\D_{m/B} x)(t)$ of the input signal.

\end{minipage}
\end{center}
\vspace*{-1mm}
\end{floatbox}

The system identification problem thus amounts to reconstructing $\mathbf{s}$ from $\vect{y} \!=\! \rev{\mathbf{X}}\mathbf{s}$, i.e., solving a linear system of equations.
Clearly, for the existence of a unique solution $\mathbf{s}$, it is necessary that the number $|\mathcal{S}|$ of unknowns be smaller than or equal to the number $N$
of equations, which corresponds to the discrete underspread condition $|\mathcal{S}|\le N$. Due to $|\mathcal{S}| = \lceil 4\tau_{\max}\nu_{\max}BD \rceil$
and $N = \lceil BD \rceil$, this is equivalent to $\lceil 4\tau_{\max}\nu_{\max}BD \rceil \le \lceil BD \rceil$ and hence, effectively, to $d_\H=4\tau_{\max} \nu_{\max} \le 1$,
which implies that only underspread systems are identifiable. Sufficiency of the underspread condition $d_\H \le 1$ for identifiability is shown by
explicitly constructing a sounding signal $x(t)$ such that ${\bf X}$ has full column rank. A viable choice for $x(t)$ is a (possibly weighted) Dirac train \cite{Kai62,sounder_tc00,kozek_identification_2005}. We have thus recovered the classical result by Kailath \cite{Kai62}, which states that a TF dispersive
system is identifiable if and only if it is underspread. Intuitively, in the overspread case, the system varies too fast
to be identifiable. A generalized
version of Kailath's result was proven in \cite{kozek_identification_2005}.
The results described above are non-parametric in that they do not impose structural assumptions on the system. Developing parametric equivalents using, e.g.,
the basis expansion model \cite{TsaGia96} is an interesting direction for further research.

The development above can be extended to systems whose spreading function support region is scattered across the delay-Doppler plane.
Such systems are identifiable if the overall support area of the spreading function is at most $1$ \cite{Bello69}.
This result holds even if the spreading function support region is not known prior to identification \cite{heckelboel12}.}

It is commonly accepted that ``good'' sounding signals $x(t)$ have a rapidly decaying temporal autocorrelation function
(see, e.g., the references in \cite{sounder_tc00}). This statement specifically applies to time-invariant systems, which induce time shifts only.
For TF dispersive systems, which cause both time and frequency shifts, our formulation of the identification problem shows that, for ${\bf X}$ in
\eqref{eq:syseqyAs} to be well-conditioned, the TF translates of the sounding signal $x(t)$ should be as orthogonal to each other as possible.
This means that the auto-ambiguity function $A_{x,x}(\tau,\nu)$ should be small for $(\tau,\nu)=(m/B,l/D)$ with $(m,l) \in \Z^2\backslash (0,0)$.

\section{Conclusion}

TF dispersive channels and WH function sets are central concepts in communications. Both are fundamentally based on the notion of TF shifts.
Our aim in this paper was to demonstrate that the corresponding TF framework is not only conceptually interesting, but also provides powerful
tools for solving problems such as pulse design in OFDM systems, characterization of the noncoherent capacity of continuous-time TF dispersive channels,
and system identification and channel estimation. Furthermore, this TF framework applies in an almost one-to-one manner
to other fields like radar and sonar (doubly spread targets \cite{vantrees3}) and quantum physics (quantization and coherent states \cite{folland89}).
We thus hope that this paper will inspire innovative research and foster cross-fertilization between the signal processing, communications, information theory,
physics, and mathematics communities.

\section*{Acknowledgments}
The authors are grateful to R.~Heckel for helpful comments and
to the authors of \cite{cfm_proc11} for permission to use the channel measurement
data in Fig.~\ref{fig:intro}.
The work of F.~Hlawatsch and G.~Matz was supported by FWF Grants S10603 and S10606, respectively.

redpajama_set_name: RedPajamaArXiv


text:
Q: How to manipulate attributes of url-less dropdown item in a the custom wordpress menu? I need to change the color of some custom link elements in dropdown of my wordpress custom menu here. There 3 items in my dropdown that dont have any url associated with them:

*

*Air Duct

*Chimney

*Dryer Vent

They will be just the headers for other dropdown items below them.
I need to be able to change their attributes such color, font-family and size, etc.
I was trying to create menu-item-type-custom and menu-item-object-custom and even menu-item-#### classes, but it didn't do it... :( Please help!

A: Enable the CSS Classes (Screen Setting -> Flag CSS Classes) and use it to customize every menu item that you want to be styled in a different manner.
You must use a Custom menu, of course.

A: Another way, using a filter if you want to do this programmatically:
add_filter('nav_menu_css_class', function($classes, $item, $args){

  if(empty($item->url))
    $classes[] = 'your-header-class';

  return $classes;

}, 10, 3);

But you should go with keatch's solution :)

redpajama_set_name: RedPajamaStackExchange


text:
Q: Z3 C API Changing Timeout at Runtime Is it possible to change the timeout value of the solver at runtime using C API?
In order to set the timeout the following can be done -
Z3_config  cfg = Z3_mk_config();
Z3_set_param_value(cfg, "SOFT_TIMEOUT", "10000") // set timeout to 10 seconds
Z3_context ctx = Z3_mk_context(cfg);
....
Z3_check_and_get_model(ctx);
....
....
Z3_check_and_get_model(ctx);

However, suppose we want to change the timeout for next query while retaining the context, is it possible to change the timeout value in between?

A: Consider moving to Z3 4.0. Z3 4.0 has new API that allows the user to create many solvers in the same Z3_context. You can set different timeouts for each solver, and update them whenever you want. Z3 4.0 also comes with a C++ layer that makes the C API much more convenient to use.
Here is an short example that sets a timeout. On my machine, Z3 will return unknown when the 1 millisecond timeout is used, and sat when the s.set(p) command is removed.
context c;
expr x = c.real_const("x");
expr y = c.real_const("y");
solver s(c);

s.add(x >= 1);
s.add(y < x + 3);

params p(c);
p.set(":timeout", static_cast<unsigned>(1)); // in milliseconds
s.set(p);

std::cout << s.check() << std::endl;


Seven young players from the Balzan FC Elite Academy are currently on a twelve-day training cap in Italy together with other young players from Sydney, Australia at the AC Milan International academy and to take part in an international tournament in Montecatini Terme.
The young players are Noah Farrugia, Andrea Grech, Brave Azzopardi, Kirsten Galea, Kubilay Karadol, Aidan Cassar and Miguel Busuttil. A member of the Balzan academy coaching staff, Omar Farrugia, joined Andrea Icardi's staff.
The contingent went first to the Novarello sport centre in Novara and then were due to play friendly matches against Monza, Novara, Alessandria and Genoa.
Former AC Milan player and current head coach of the AC Milan Academy Andrea Icardi said he was glad to help these young players from the Balzan academy in trying to fulfill their dream of becoming professional players. Balzan FC president Anton Tagliaferro said that thanks to such experiences, young players from the academy are getting the opportunity to live the life of a professional player for a few days. he said he hopes the young Balzan players make the most out of this adventure while training for one of the top academies in the world.


text:
— Top Menu —ENGLISH ESPAÑOL
— Main Menu —National>Entertainment National>Local Economics National>Local Politics National>Society National>Sports
National>Entertainment
National>Local Economics
National>Local Politics
National>Society
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La Cumbre is still being monitored on Fernandina Island
Debuts the third airline that offers direct flights to New York in the country
Non-traditional destinations boosted banana shipments
Rafael Correa commanded in a criminal organization, according to elements of conviction highlighted by Judge Daniella Camacho
Community tourism makes its way with local initiatives
Third-party properties may be finished off with COIP reforms
These are the most profitable companies in the Ecuadorian stock market
Correistic assembly members who were at the Mexican embassy in Quito received help to leave the country
Dollarization brings economic stability to the Ecuador
The dollar pushed exporters not to rely on devaluations
Ministry proposes reducing tariffs for telephony
Ecuadornews:
The policy of the Telecommunications portfolio prioritizes the universal access of the population to ICT and for this the sector ensures that state incentives are required.
The tariff burden is one of the factors that must be overcome to universalize access to telephony and mobile internet services, according to the Minister of Telecommunications, Guillermo León.
"The universal access of the population to new communication technologies is restricted at the moment when, through tariffs, the cost of terminals (mobile phones) is increased, discouraging and slowing down access to better technology," he said this Wednesday the Secretary of State, in the event of presentation of the Strategic Vision 2018 – 2021 of the Telecommunications Sector. In this sense, Minister León stressed the need for the articulation of a public policy to favor the reduction of tariff charges.
Francisco Balarezo, executive director of the Association of Internet Service Providers (Aeprovi), said that it is not feasible that cell phones entering the country have a tax of 15% for special consumption, when they are currently essential basic supplies for the development of society.
"A review must be made of the tax charges that have an impact. As an industry we believe that the priority is to have access to technologies because that moves the economy and reduces the digital gaps, in addition to improving the opportunities in the market, "said the official.
With this same criterion, the Association of Telecommunications Companies (Asetel) agrees, which considers necessary a public policy focused on eliminating specific taxes for public telecommunications services such as the tax on the exit of foreign currency (ISD), as well as the tax on special consumptions.
According to Asetel, access to telecommunications services is still restricted for the most vulnerable segments of the population.
Four axes make up the telecommunications vision until 2021
In order to strengthen the foundations of the sector and move towards the universalization of access to new information and communication technologies (ICT), the Ministry of the industry based its strategic vision until 2021 on four axes with a strong focus on the productive sector.
The first axis is connectivity, which seeks to promote the deployment of infrastructure to expand the coverage of telecommunications services in underserved areas and stimulate migration to high-speed networks.
Another axis is the orientation towards an electronic government that involves, among other things, the reduction of procedures and regulations.
The Ministry also values the security of information and the protection of personal data, and finally, as the fourth axis, there is economic development through digital transformation.
Promotion of connectivity
To promote connectivity, the regulation of concession rights and tariffs for the use of spectrum will be updated, as well as the promotion of regulatory incentives and the reduction of tax charges to terminals.
66 percent is the goal for the population penetration of smartphones until 2021.
Regional reference
One of the goals of the sectoral strategic vision is for Ecuador to be recognized in 2021 as a regional reference. (I)
Source: https://www.eltelegrafo.com.ec/noticias/economia/4/telefonia-movil-aranceles-ecuador
Inter-American Court will decide if the communication is a public service in Ecuador
Ecuadorian dreamed big and triumphed with a protein supplement in the US
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redpajama_set_name: RedPajamaCommonCrawl


This has been a very odd week or two. We are starting to feel like we are in a time warp or something. Our stay at the lake started out wonderful. A nice lake, in a nice place. Quite, pretty sunsets, swimming, all the good things.
Then Friday we woke to a giant hatch of mutant bugs. Like a bajillion of them. Thankfully they don't bite, that would have been really bad as I react horribly to bug bites. They cling to the shade side of the RV, begging to get in. And quite a few do get in. Yuck.
That morning we braved the bugs and walked to an old cemetery. It was an odd mix of very old, and new headstones. There were some from the late 1800's. There was also a fairly large headstone with a chain fence around it. The ornamentation on the fence was very cool. I would love to know if these were strictly ornamental, or did they have a certain purpose.
The previous night we went exploring a small homestead that was just up the road. We always wonder, who lived here, what was their life like, when exactly did they live here.
And it pays to be observant I guess. I was walking the dogs down the dirt road, and glanced toward the barbed wire fence. This heart caught my eye.
Saturday was really hot, even more bugs, and we were hardly able to be outside. Yesterday we moved on to an RV park in Three Forks. I tried to see if we could just go to Bozeman but the RV park doesn't have room until Wednesday. When it gets really hot, boondocking can be hard unless you can get under a tree or something. The heat can be unbearable, with no power there is no A/C.
We are seeing more of Montana than we intended. I am still withholding final judgement until after the Bozeman art show this weekend. But I will say, we are pretty done with being here.
Honestly, Mary, it doesn't sound like I would be very happy with that place either! Between the bugs and the no A/C I would be ready to leave!! Hope things improve SOON!!
You find beauty EVERYWHERE. That heart….old, bent, rusted, clad in spider silk. Yes, you find beauty, anywhere.
Thanks, I look for it everywhere.
Hope things get more comfortable fast!
More comfy now we have shade, and power. And that's perfect, beauty out of pain.
Hello, gorgeous sky capture! I love the shot of the heart barbed wire. I do not like bugs either, UGH! Good luck with the art show! Happy Monday, enjoy your new week!
Whew! I bet it is like an oven with no a/c out in the sunshine! We have a Japanese Beetle infestation which is disgusting, not fans of bugs here either! Hope your art show goes well and you can move on to better places.
You are so welcome, and I'm glad you like them.
Intriguing photos of that one-time homestead, and the first sunset photo is absolutely magnificent. I wouldn't like being surrounded by swarms of bugs…yuck! I never thought about the fact that you wouldn't have any air conditioning when boondocking, I always thought you had a generator and could park just about anywhere. It's so interesting to hear about both sides of RV living!
We have solar power, and a generator. We tried the generator for a little while, but it got too hot and Al was afraid we would burn the house down.
I've been thinking about you, and wondering how you are, and where you are. Even us travelers have our favorite places.
It doesn't sound like Montana is getting any better for you! I hope the show at least goes well so it's more worth it. Then I imagine you'll hightail it out of there and never look back!
You don't have a generator for your RV so you can at least have some power? I'm not sure I could rough it quite that much!
We have solar power, and a generator. Solar gives us all the power we need, but you can't install enough solar to run A/C. We have a generator too, and ran it for a little while. But it heated up pretty good, and Al was afraid we would burn the house down.
Better to suffer with some heat than lose your whole house! We had storms recently and a lot of people lost power, on a very hot night. My sister told me their generator wouldn't run their A/C either, even a big whole house one like they have. I'm glad you have some power at least though, that has to make your life easier.
We will send good vibes that the art show is well worth the trouble you've been having there!
Hoping the art show goes really well for you!! Sorry Montana has been such a "bust" for you guys; but at least you got some great photos out of the trip! And, hey, as my dad would have said "if you don't try, you'll never know". These photos are intriguing, each in their own way. The heart in the barbed wire is so "cool"!! And my mind would have been wandering all over the place at the old homestead. ?
BUGS. I hate bugs. I'm already done with the bugs this season and I bet we still have at least two more months of them. It seems like every night I am killing a bug in my room. Poor Sampson, he hears me exclaim, "Jesus Christ" and he jumps up before the swatting begins.
Last week I had a grasshopper (A GRASSHOPPER) touch my forehead in passing. It landed on my wall. A grasshopper at 10:30 at night. Only me.


This year, Anthony's and my family came over for Christmas. We had the best time together. It's one of those memories that you want to frame.
Every since I was a child I have had short term memory loss. I do remember most things but there are gaps of memory missing. I did not know what this was until someone discovered it for me.
Dysnomia is a type of memory disorder where people sometimes have difficulty recalling words and names, and situations. It is a form of nominal aphasia, a term used to refer to conditions where people cannot remember specific words.
It is not, but only a small type of Anomic aphasia (anomia) is a type of aphasia characterized by problems recalling words or names. Subjects often use circumlocutions (speaking in a roundabout way) in order to express a certain word for which they cannot remember the name.
My husband says I have a GIFT. I forget a lot of a neg. things. I will still remember what happens, but cant recall the emotion over time. Pretty soon I forget the whole thing all together. So, If we fight I will soon forget what we were even fighting about. When people have been away from me that I really didn't know for a long time. I forget their names and who they are.
Recently, our church went through a split. It took a lot of tares and wheat alike. Yes, I may have ranted for a day or so. But because I didn't know them for very long. I have already forgotten some of their names. The emotion is gone. It's a gift! That's why I guess I am a pretty positive person. I give the neg. to God! I have total faith that everything is gonna be alright! God's got this! I trust him. I have had to all my life depend on my heavenly father who has always taken care of me. My Abba Daddy!
Sometimes after a long hard season of Battle, you just want to lift your hands to your heavenly father with much gratitude, praise and shouting Thank You God! Thank you for keeping your hands over me while the enemy would like to me bring down. Today, I stand and say Yes, I am still here. Still worshiping my savior who is me complete defense.


text:
Advances in Dermatology and Allergology/Postępy Dermatologii i Alergologii Advances in Interventional Cardiology/Postępy w Kardiologii Interwencyjnej Archives of Medical Science Biology of Sport Central European Journal of Immunology Folia Neuropathologica Journal of Contemporary Brachytherapy Polish Journal of Pathology Prenatal Cardiology Videosurgery and Other Miniinvasive Techniques
Folia Neuropathologica
Current issue Archive Manuscripts accepted About the journal Journal's reviewers Abstracting and indexing Subscription Contact Instructions for authors
Editorial System - Submit your Manuscript
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APOE genetic variants and apoE, miR-107 and miR-650 levels in Alzheimer's disease
Michal Prendecki, Jolanta Florczak-Wyspianska, Marta Kowalska, Jan Ilkowski, Teresa Grzelak, Katarzyna Bialas, Wojciech Kozubski, Jolanta Dorszewska
Folia Neuropathol 2019; 57 (2): 106-116
DOI: https://doi.org/10.5114/fn.2019.84828
Online publish date: 2019/05/14
JabRef, Mendeley
Papers, Reference Manager, RefWorks, Zotero
Prendecki M, Florczak-Wyspianska J, Kowalska M, et al. APOE genetic variants and apoE, miR-107 and miR-650 levels in Alzheimer's disease. Folia Neuropathologica. 2019;57(2):106-116. doi:10.5114/fn.2019.84828.
Prendecki, M., Florczak-Wyspianska, J., Kowalska, M., Ilkowski, J., Grzelak, T., & Bialas, K. et al. (2019). APOE genetic variants and apoE, miR-107 and miR-650 levels in Alzheimer's disease. Folia Neuropathologica, 57(2), 106-116. https://doi.org/10.5114/fn.2019.84828
Prendecki, Michal, Jolanta Florczak-Wyspianska, Marta Kowalska, Jan Ilkowski, Teresa Grzelak, Katarzyna Bialas, and Wojciech Kozubski et al. 2019. "APOE genetic variants and apoE, miR-107 and miR-650 levels in Alzheimer's disease". Folia Neuropathologica 57 (2): 106-116. doi:10.5114/fn.2019.84828.
Prendecki, M., Florczak-Wyspianska, J., Kowalska, M., Ilkowski, J., Grzelak, T., Bialas, K., Kozubski, W., and Dorszewska, J. (2019). APOE genetic variants and apoE, miR-107 and miR-650 levels in Alzheimer's disease. Folia Neuropathologica, 57(2), pp.106-116. https://doi.org/10.5114/fn.2019.84828
Prendecki, Michal et al. "APOE genetic variants and apoE, miR-107 and miR-650 levels in Alzheimer's disease." Folia Neuropathologica, vol. 57, no. 2, 2019, pp. 106-116. doi:10.5114/fn.2019.84828.
Prendecki M, Florczak-Wyspianska J, Kowalska M, Ilkowski J, Grzelak T, Bialas K et al. APOE genetic variants and apoE, miR-107 and miR-650 levels in Alzheimer's disease. Folia Neuropathologica. 2019;57(2):106-116. doi:10.5114/fn.2019.84828.
Alzheimer's disease (AD) is a progressive neurodegenerative dementia in adults. Pathogenesis of AD depends on various factors, including APOE genetic variants, apolipoprotein E (apoE) phenotype and oxidative stress, which may promote both DNA and RNA damage, including non-coding RNA (ncRNA). Among ncRNAs, microRNA (miRNA) is known to contribute to pathologic processes in AD.
The aim of the study was to analyse the plasma concentration of apoE by ELISA as well as the plasma levels of miR-107 and miR-650 by qPCR in relation to APOE genetic variants and clinical features including the age of onset and dementia severity in 64 AD patients and 132 controls.
Our data showed that a low apoE plasma concentration was a risk factor for developing AD (OR = 5.18, p = 6.58E-06) and was particularly pronounced in severe dementia (p < 0.001) and correlated with cognitive functions (R = 0.295, p = 0.020), similarly as the level of miR-650 (R = 0.385, p = 0.033).
The presence of APOE E4 allele in both AD patients and controls led to a reduction in apoE, while APOE E3/E3 genotype was associated with an increased apoE concentration and level of miR-107 in AD (p < 0.05) which was inversely correlated with the number of APOE E4 alleles (R = –0.448, p = 0.009). Additionally, patients with the onset at 60-69 years of age showed a reduced level of miR-107 (p < 0.05, as compared to AD above 80 years of age).
Changed levels of plasma apoE, miR-107 and miR-650 may be a marker of the neurodegenerative process in the course of AD, associated with amyloid β metabolism and inordinate cell cycle.
APOE, apoE, miRNA, miR-107, miR-650, Alzheimer's disease
Leksykon genetyki w psychiatrii
prof. dr hab. n. med. Joanna Hauser, dr n. med. Monika Dmitrzak-Węglarz
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Successfully Added Tie Camera with MP4 Player 512MB, 2.5-inch LCD, 1.3M Pixel to your Shopping Cart.
The World's Smallest Color Pinhole Camera built-in the fashionable Tie. When tie this tie with your clothes lets you record exactly what you are looking at. No external power needed, just connect it directly with MP4 player to hidden recording so you will not miss any action. It is also great as a surveillance camera, a toy for hobby enthusiasts, as body worn solutions and so on..


Application of Maximum Diametric Length Measure for Space Syntax etc..
Re: Application of Maximum Diametric Length Measure for Space Syntax etc..
first stage result of Mass Observations !
Re: first stage result of Mass Observations !
interesting research tool of the day.
Re: interesting research tool of the day.
Urgent, pedestrian modelling of cities.
Re: Urgent, pedestrian modelling of cities.


text:
Xbox Series S: this funny project turns the machine into a portable console
Your Xbox Series S could soon become the best portable console in the world thanks to a new Kickstarter. This crazy project aims to provide a screen to the console, so you can play it wherever you are
The "xScreen" is an attachable device that owners can connect to their Xbox Series S and use as a screen. It has a 1080p definition and a 60 Hz refresh rate. This means that it will not be possible to enjoy some games in 4K and 120 frames per second. The device is also equipped with built-in speakers. To say the least, the resemblance to a laptop is striking.
The most interesting, of course, is not the technical details of the screen, but its main advantage: to be able to turn an Xbox Series S into a portable console. At the beginning of the year, we explained why Microsoft never released a portable version. It should also be noted that since it is only possible to buy games in digital format, you will not have to carry all your game boxes when you travel.
WHEN WILL THE KICKSTARTER PROJECT BE LAUNCHED?
The Kickstarter has not yet been launched, but it already seems to be well received by Internet users. However, the project is not completely original, as it is not the first screen of its kind. Indeed, this kind of devices already existed for some consoles such as the GameCube. In 2018, a modder also managed to turn the Xbox One X into a portable console for 2,000 euros.
The xScreen Kickstarter is scheduled to launch soon, but anyone interested can already book the device for a symbolic dollar on the xScreen website, which will lock in its $159 early bird price for the first 200 units. The next 200 will go up to 189 dollars, while the final price will be much higher. It will cost 249 euros for the screen.
While waiting for the xScreen to be released, you should know that Xbox consoles are also available in a cloud thanks to xCloud. Yesterday, we even learned that it would soon be possible to play Xbox on any smart TV, even without having a console.
The First Commercial Game and the First Home Console
Xbox BEATS Sony's PlayStation 5 REVEAL :Xbox Series X Has HUGE Advantage VS Ps5
PS5: a new controller for the PSVR 2
PS5: STORE INVENTORY REPLENISHED EARLY MARCH. WHERE TO BUY CONSOLE


Below are the secondary essay prompts for the University of Vermont College of Medicine in Burlington, VT.
The Larner College of Medicine at the University of Vermont strongly values diversity. Larner COM recognizes that diversity extends beyond chosen and unchosen identities, and encompasses the entirety of an individual's experiences. These experiences embody many dimensions that include (but are not limited to) the personal, familial, geographic, intellectual, economic, professional, linguistic, and societal domains. Thus, each and every member of the Larner COM community brings a unique perspective to a shared goal of training compassionate and competent physicians.
With this in mind, please reflect on your own diverse experiences and how those will allow you to successfully participate in teams and engage in active learning with your peers at the Larner COM. When composing your response please consider how these experiences impacted your understanding of others and yourself as well as any new insights gained. (500 words max).
Below are the secondary essay prompts for the University of Vermont College of Medicine.
The University of Vermont College of Medicine values diversity as a driver of excellence. We actively seek diversity and inclusion within our academic community of students, faculty, and staff, as well as those we serve through teaching, research, patient care and community engagement. We are committed to building an educational community made up of members with diverse talents, experiences, opinions, and backgrounds.
Please write a short response to one of the following prompts. For either question, when composing your essay, please consider the personal growth you experienced, an impact you may have had on others or a transformation you may have experienced (3000 characters maximum).
In your journey towards a career in medicine, what experience do you think most convinced you that you want to serve others as a physician?
There are many different types of teams. Effective teams are composed of people with diverse life experiences and varying viewpoints. Reflect on an experience you have had working on a team.
Need more help on your 2018-2019 University of Vermont College of Medicine secondary essays? Click here to sign up for a FREE 15 minute consultation.


text:
January 20, 2021 Guns In The News
Silencer Central Announces Direct-to-Consumer Shipping
Why You Should Use a Shotgun Sling for Personal Defense
New for 2021: Springfield Armory XD-S Mod.2 OSP Pistol
New for 2021: FK Brno PSD Multi-Caliber Pistol
Watts Wants Credit Card Companies to Monitor Gun Parts Purchases?
Home»Politics»Mike Bloomberg-Moms Push Gun-Control By Putting Good People at Risk ~ VIDEOS
Mike Bloomberg-Moms Push Gun-Control By Putting Good People at Risk ~ VIDEOS
Guns In The News 2020-10-12
Guns In The News / October 12, 2020 /	444
Article first appeared on Ammoland.com
USA – Who are the real Bad Guys here?
The right to keep and bear arms is the one right the radical, anti-gun Left doesn't want you to have because as long as We the People have power, theirs is limited.
So when groups like Moms Demand Action, a "volunteer" wing of Everytown for Gun Safety try to disarm you while supporting the release of violent felons onto the streets of our neighborhoods, when they push for gun-free school zones that put children and teachers in danger by removing their security, when they push for gun restrictions that put good people in danger by limiting their ability to defend themselves in their own homes, and when they're funded by someone who pays off the debts of convicted felons so they can control their votes, you have to ask yourself; who are the real bad guys here?
Michael Bloomberg's own 2020 Presidential campaign website revealed his plan to "end cash bail." In other words, put criminals back on the streets by removing the one thing that keeps them in jail prior to trial. In support of Bloomberg and his policies, Moms Demand apparently gave his campaign their "members email list" (it would appear, without consent) so the Bloomberg campaign could seek the votes of MDA's members. According to Mother Jones, Aimee Tavares, a senior national organizing director for Moms Demand Action, said,
"Given his unique role in launching Everytown, his leadership in advancing the cause of gun safety, and the significant investments he's made in helping to grow our movement, Mayor Bloomberg's campaign is temporarily renting access to our list of supporters so they can communicate his plans for achieving our shared goal of a nation free from gun violence,"
Moms Demand also opposed Missouri House Bill 1961; "Keep Our Schools Safe Act," claiming that the bill would increase the risk of "gun-violence." In fact, the bill would provide additional security for students and teachers by allowing qualified School Protection Officers the ability to save lives should the need arise. We saw what happened in Parkland when security was limited, but it appears that the gun-grabbers like it that way. Contrary to MDA's claims, Joe Biden's 1990 Gun Free School Zones Act has produced a significant increase in school killings by forcing children and teachers into unarmed helplessness. The article on the MDA website condemning HB1961 is riddled with typical anti-gun fear-mongering statements in pursuit of their political agenda and unarmed schools.
From the Moms Demand Website:
…"increasing the risk of unintentional shootings and gun violence."
…"lawmakers are instead taking us backwards and advancing risky bills that would put children, teachers, and school officials in danger."
…"puts children at risk of unintentional shootings and escalating conflict."
Everytown and Moms Demand gladly support and take funds from the one guy who was bold enough to attempt to publicly, buy the black vote for Joe Biden in Florida. With over $16m raised, the former NYC Mayor planned to pay off the debts of released prisoners in exchange for their votes.
According to the Washington Post, in a memo, Michael Bloomberg said, "We have identified a significant vote share that requires a nominal investment," the memo read. "The data shows that in Florida, Black voters are a unique universe unlike any other voting bloc, where the Democratic support rate tends to be 90%-95%."
No representatives from either Moms Demand or Everytown have publicly denounced this behavior.
While American cities are burning to the ground, businesses are being looted and destroyed, and good people are being viciously attacked, some Mayors refuse to stop the violence. During this chaos, groups like Moms Demand Action are trying to convince you that being unarmed and helpless in this environment is a good condition to be in.
Smart people are no longer falling for their lies. We see this in the massive increase in gun sales across America. Despite the devious and sinister messaging coming from these anti-rights lobbying groups, real Americans are able to see just how important it is to be able to defend ourselves and our families.
The Second Amendment was written because our Founding Fathers saw these dangerous, anti-Americans, and the tyrants they support coming over 200 years away. The Second Amendment is not a privilege; it's your right, and the gun-grabbers hate that.
The 2nd Amendment is not a privilege. It's your right.
About Dan Wos, Author – Good Gun Bad Guy
Dan Wos is a nationally recognized 2nd Amendment advocate and Author of the "GOOD GUN BAD GUY" series. He speaks at events, is a contributing writer for many publications, and can be found on radio stations across the country. Dan has been a guest on the Sean Hannity Show, NRATV, and several others. Speaking on behalf of gun-rights, Dan exposes the strategies of the anti-gun crowd and explains their mission to disarm law-abiding American gun-owners.
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THE YEAR THE SECOND AMENDMENT STRIKES BACK
Man Defends Himself As He's Encircled By Gang Of Armed Men - Takes One Of Their Guns, Kills Attacker, Others Flee
Guns In The News
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South Carolina Attorney General Sues City Over Anti-Gun Ordinances


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The Hüfi Glacier () is a  long glacier (2005) situated in the Glarus Alps in the canton of Uri in Switzerland. In 1973 it had an area of .

See also
List of glaciers in Switzerland
List of glaciers
Retreat of glaciers since 1850
Swiss Alps

External links
Swiss glacier monitoring network

Glaciers of the canton of Uri
Glaciers of the Alps

redpajama_set_name: RedPajamaWikipedia


text:
Treacherous Transparencies (GER ED.)
Thoughts and Observations Triggered by a Visit to Farnsworth House
Treacherous Transparencies analyzes transparency as expressed in architecture and art in an attempt to understand the intentions and objectives that underlie its use by pertinent architects and artists.
The publication looks at a few important works by selected artists and architects who work with transparency as an artistic strategy, which they implement primarily by using glass and mirrors but other media as well. The architects and artists listed together in this context form an unlikely alliance: Bruno Taut, Ivan Leonidov, Marcel Duchamp, Mies van der Rohe, Dan Graham, and Gerhard Richter. But they do have something in common: their work marks salient way stations in the story of modernism up to the present day.
Published in the context of the inaugural Mies Crown Hall Americas Prize (MCHAP). Launch of the publication series by the inaugural MCHAP award winners Jacques Herzog and Pierre de Meuron for their project 1111 Lincoln Road in Miami Beach.
See Preview on issuu
Author: Jacques Herzog & Pierre de Meuron
Illustrations: Color & Black and White
Cover: Hardcover
ISBN: German 978-1-945150-12-8
Price: 22€ /$24.95 / £18
previous post: Treacherous Transparencies (ENG ED.)


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5 reasons you still need a dedicated audio player in 2016 - and some reasons you don't
By Jon Porter 2016-08-26T11:00:00.238Z
DAP to the future
There was a time, around that of the iPod Mini, where almost everyone I knew would refuse to leave the house without a music player tucked into their pocket.
I was no different. My trusty Sony Walkman-branded MP3 player accompanied me everywhere, and even though its internal storage seems puny by today's standards it felt at the time as though it contained all the music I could possibly want to listen to.
Personally, I stopped using an MP3 player (or digital audio player / DAP as they're referred to in audiophile circles now) when I got my first smartphone. It might not have had the slickest interface, but it felt so much more convenient than carrying around two devices.
Apple still uses the iPod name alongside its Touch, Nano and Shuffle models, but it discontinued the iPod Classic in late 2014 to focus on these smaller form-factor devices.
Recently though there have been some interesting developments in the world of DAPs that mean that you might want to give them a look once again if you're looking to make the most of your portable listening.
Here's five reasons why you may want that second device back in your pocket...
1. They'll support your high-end headphones
Once you've delved into the world of high-end headphones, it's unlikely that your phone will be able to support their more advanced features.
There aren't any smartphones out there that support balanced headphone output, which offers an enhanced level of sound quality over their non-balanced equivalents, and Apple continues to refuse to build Bluetooth AptX support into its iPhones.
Meanwhile many DAPs, such as Astell&Kern's AK70, will include support for these features as standard.
Going beyond these features, high-impedance headphones require much more power to be driven properly. Most phones are unable to do this, but it's perfectly possible to find digital audio players, such as HiFiMan's HM-90I, that can drive them without issue.
...which aren't designed for portable listening
There's a reason why most phones don't support high-end headphones. These headphones are bulky, and aren't designed for portable listening in the first place.
When it comes to open-back headphones you might find it impossible to listen to them while outside anyway because of the sheer amount of noise that leaks into them.
Of course you might use a portable music player in a completely silent office where these concerns won't affect you, but the likelihood is that the reason you want a portable music player is to listen to it when properly out and about, not sitting at a desk.
2. It'll have dedicated music controls
Most phones nowadays are stripping away as many of their buttons as possible in an attempt to be as sleekly designed as possible, but physical buttons are very handy when you're listening to music on the go.
Whether it's skipping tracks or just pausing them quickly to ask someone a question, dedicated buttons -like those found, again, on Astell&Kern's AK70 - can be very helpful.
Yes these are becoming more common with the proliferation of headphone in-line remotes, but if you don't want to upgrade your existing pair of headphones then you need some buttons on the device to make your life easier.
3. Its storage is dedicated to music...
Uncompressed audio takes up a lot of space, and high resolution audio takes up more space still.
If you've got a phone that's filled with the latest apps, games, and all those slow-motion videos that you've been meaning to back up and delete then it can be hard to find some space for your music collection.
A dedicated music player has storage that's reserved for music alone, and as such you can wave goodbye to having to micro-manage your folders to make room for your newest 24-bit find.
...but many phones offer expandable storage
Of course your storage problem can be massively reduced if you opt to go with a phone with support for external storage options in the first place.
MicroSD cards are getting cheaper and larger with each passing year, and a sensible phone purchase means that you can upgrade its storage to your heart's desire.
You'll still need to balance your music storage needs with all your other apps, but if you use your internal and external storage sensibly then there's much less cause for concern.
4. It'll support hi-res audio…
The situation has improved over recent years with the introduction of high resolution DACs into smartphones, but the vast majority of the devices are still limited to CD-quality audio.
That might not bother you if you're happy with the level of music quality that CDs provide, but high-resolution music can offer you a level of music quality that's often quite surprising.
Beyond high-res audio, MQA is the next big development in audio quality, and as it currently stands there isn't a phone in the world that supports the standard. Meanwhile Pioneer has already released a DAP that can play MQA files, the XDP-100R.
Hi-res audio might not offer the same leap in quality as we saw between SD and HD video, but it's a substantial improvement that can really sing when paired with the right headphones.
...but with the right accessories so can your phone
Your phone's internal DAC might not be able to support hi-res audio, but with the addition of an external DAC it might be easier than you think to get it upgraded.
The exact accessories that are compatible will vary from phone to phone, but using the iPhone as an example, you can buy a lightening to USB adaptor which will allow you to plug in a DAC such as AudioQuest's Dragonfly Red which supports hi-res music up to 32-bit.
Then it's just a case of making sure the music playing software you're using in the phone supports the correct resolution of file and you should be ready to go.
It's not quite as neat a solution as having everything built into a single DAP, but it might be an option if you don't want to carry around two devices.
For more information, check out our guide to using an external DAC with an iPhone.
5. It's not going to drain your phone's battery
On long flights and plane journeys I cannot live without music. Sure I might try and read a book, or play a game or two, but the thing I'm going to return to time and time again is my music, which I can listen to no matter how travel weary I am.
But if I'm on a long journey then there's a high chance that I'm going to need to use my phone at the other end, be it to phone the person I'm meeting or else find my way using Google Maps or CityMapper.
This can often leave me conflicted. I want to listen to music, but I need to keep an eye on my phone's battery lest it die before I reach my destination.
A DAP is great in this regard, because I know that no matter what happens my phone battery is safe. I can listen to music until my audio player's battery completely dies and still have enough juice left in my phone to order a cab at the other end.
It means that you can enjoy your music without having to worry or ration, and that can make all the difference on a stressful journey.
Conclusion: There's still a time and a place for a DAP
It's unlikely that we'll ever see a return to the days of the original iPod where everyone carried their trusty MP3 player with them at all times.
But hopefully this list has proved that depending on how and where you listen to music that there still might be a reason to invest in a dedicated digital music player.
See more Portable audio news


Divorces can be hard. I always knew this, but did not realize how hard until I was in the middle of mine. I went from being married one day to fighting with my spouse about who got to spend more time with the kids. My confidante and friend was acting like my enemy. When the divorce proceedings started, I had no idea it would be so complicated. There was so much information out there and a lot of it was conflicting. I started this blog for other people who are just as confused and who need real advice from someone who has been through the process.
A bankruptcy attorney is a fantastic financial resource that can help you deal with any debt that might be too overwhelming for you to pay off or that is putting an undue hardship on you and your finances. Listed below are two reasons to speak to a bankruptcy attorney.
One of the most irritating and frightening parts about being deep in debt is the fact that some bill collectors are going to be very aggressive when attempting to collect on the debt that you owe. While the majority of bill collectors are going to be quite polite, there are going to be a few that are not only going to be more than a little aggressive and that will not stop contacting you all the time but that will also resort to attempting to frighten or bully you into paying your debts, even if you don't have the finances at the moment.
However, the way a bankruptcy attorney can help you deal with these individuals is by providing a buffer between you and the bill collectors that you don't have to deal with them directly. The reason for this is that once a bankruptcy attorney has filed your paperwork, the bill collectors are no longer able to garnish your wages, seizure property, or contact you in any way during the bankruptcy process. If they need to speak to anyone about any part of your debt, they must speak to your lawyer or the bankruptcy court or face substantial fines and penalties.
Another reason to speak to a bankruptcy attorney is the fact that he or she can provide you with the opportunity to start over. The reason for this is that the most common form of bankruptcy, a Chapter 7 bankruptcy, is one that will wipe way the vast majority of your debts completely.
This means that you are no longer have to figure out a way to pay those debts, but you also will have to take a substantial hit to your credit score and surrender some forms of property in order to qualify for the discharging of your debt. While that may sound a bit difficult to deal with, it is more than worth it if you have a massive amount of debt that you cannot conceivably pay for in the near future or if you and your family are struggling to meet your basic needs due to the amount of money that you have to pay every month to your creditors.
Make an appointment with a bankruptcy attorney today in order to discuss the various ways in which he or she may be able to assist you with your debts. A bankruptcy attorney can help you out by getting the bill collectors to stop harassing you and giving you the opportunity to start over. For more information, visit a website such as http://www.tblakelaw.com.


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The 2018-2019 All Southwest Conference girl basketball teams have been released by the conference. Broken Bow's Kali Staples was selected as SWC girls player of the year. She is joined on the all conference first team by Broken Bow teammate Majesta Valasek as well as Mackenzie Smith (McCook), Jaedy Commins (Ogallala), and Liviya Wier (McCook).