text:
\section{Introduction}
\label{S1}

Underwater acoustic (UW-A) communication has been developed for various applications, including offshore remote control, undersea pollution control, environmental monitoring, underwater communication and sensor networks,  surveillance and defense \cite{melodia13,demirors15,akyildiz04,stojanovic96}.

Currently, there are still many challenges in UW-A channels, such as severe attenuation loss, multipath propagation delay, ambient noise, Doppler effect, and frequency selective fading that restrict our technology from advancing significantly \cite{melodia13}.

Chirp signals have been utilized widely in radar and sonar applications \cite{fitzgerald74,lurton02} primarily due to their resilience to multipath propagation delays and Doppler spread as well as superior correlation characteristics. Applications in UW-A communications include linear chirp signals for navigation and underwater probing \cite{austin94,chiu15}, packet synchronization \cite{sharif00,kilfoyle00}, channel estimation \cite{wu12}, reliable UW-A communications \cite{he09}, Doppler estimation and compensation \cite{lurton02,sharif00}, and robust feedback communications \cite{demirors14}.

However, linear chirps are not orthogonal rigorously \cite{springer00}. To tackle this problem, some researchers adopt various categories of orthogonal chirp transforms, such as fractional Fourier transform \cite{en14,solyman12}, and Fresnel transform \cite{ouyang16} for supporting multicarrier communications and high data rates. However, the disadvantages of these chirp transforms are high weighted computation and complex implementations. Hence, there is of interests to design orthogonal chirp waveforms and low complexity receivers in UW-A environments \cite{huang17}.

Propelled by the need for attaining higher data rates and multiple users, chirp signals have been utilized in multicarrier communications. Some researchers applied chirps in OFDM for enhancing Doppler spread estimation, and delay tolerance \cite{dida16,barbarossa01}. Some focus on waveform designs for multiple transmitters in radar systems \cite{kim13}, orthogonal chirp signals in multiple input multiple output (MIMO) systems \cite{wang15}, and CSS for enhancing orthogonality of transmission of different users \cite{yang11,cheng15}. Several works are built on different categories of chirp bases in multicarrier systems, e.g., fractional Fourier transform (FrFT), fractional cosine transform, and Fresnel chirp transform \cite{en04,solyman12,attar17,ouyang16}. On the other hand, chirps can be exploited in multiuser communications for multiple access \cite{shen06_div}. Multicode scheme can enable simultaneous transmission of multiple chirp waveforms \cite{liu06}. However, these chirp transforms are of high computational overhead and complex implementations. Hence, developing an efficient and low complexity multicarrier chirp system is still of interests and researches in RF wireless communications.

In this paper, we consider a single-input single-output (SISO) UW-A communications that information symbols are carried over orthogonal chirp frequency subcarriers in adaptive multiuser multicarrier chirp-division multiplexing (MU-MCDM) systems. The orthogonality of chirp waveforms are analyzed by their cross-correlation coefficients. Both continuous orthogonal chirp transform (OCT) and discrete orthogonal chirp transform (DOCT) are introduced for time-domain or digital implementations. The system is adaptive in preamble duration, number of subcarriers, portion of pilots, guard interval, and multiuser scenario to optimize performance of transmission in variant underwater environments. Moreover, we develop a low complexity coherent receiver of channel estimation and maximum-likelihood (ML) symbol detection. On the other hand, we discuss the relationships of OCT and DOCT and their various applications. In addition, MCDM system is compatible with prevailing OFDM architectures. Then, simulation studies are demonstrated over UW-A multipath channels for evaluating adaptive MU-MCDM systems.

\vspace{2pt}
The rest of the paper is organized as follows. Section \ref{S2} addresses OCT representations. System Model is described in Section. \ref{S3}. Receiver design of the ML coherent receiver is introduced in Section \ref{S4}. Discussion of proposed approaches are illustrated in Section \ref{S5}. Simulation results are demonstrated in Section \ref{S6}. Finally, several concluding remarks are drawn in Section \ref{S7}.

\section{Orthogonal Chirp Waveform Design}
\label{S2}
\subsection{Chirp Waveform Design}
A linear chirp signal is considered, where waveform's frequency change linearly over time. The $m$-th chirp waveform can be represented as
\begin{align}
\psi_m(t) =  \sqrt{\frac{1}{T}} e^{j(2\pi m\Delta ft+\pi \mu t^2)},~0\leq t\leq T
\label{psi}
\end{align}
where $T$ is the symbol duration, $\Delta f$ the frequency spacing between chirps, $\mu\overset{\Delta}{=}\frac{B_c}{T}$ the chirp rate, $B_c$ the bandwidth of a chirp signal, and the waveform's energy is normalized to 1. $m\Delta f$ is the initial frequency of the $m$-th chirp waveform. For the chirp rate, $\mu >$ 0 is a up-chirp signal (frequency growing with time), while $\mu <$ 0 is a down-chirp signal (frequency reducing with time).

Proposed chirp signals are complex, comprising both in-phase and quadrature-phase parts, so as we process channel estimation, channel coefficients should be treated as complex representations. Moreover, additional bandwidth assigned to chirp signals can support a degree of freedom for resilience of multipath effect and channel fading. Then, we examine the orthogonality of chirp waveforms in derivations.

\subsection{Orthogonality of Chirp Waveforms}
Characteristics of orthogonality of chirp waveforms are crucial in our waveform designs, so we analyze the orthogonality of chirp waveforms with cross-correlation coefficients. Then, the cross-correlation coefficient of the $k$-th and the $l$-th chirp waveforms can be derived as
\begin{align}
\begin{aligned}
\rho_{kl}  &= \int_0^{T} \psi_{k}(t) \psi_{l}^*(t)dt\\

      &= \frac{1}{T} \int_0^{T} e^{j2\pi (k-l)\Delta ft}dt\\
      &= \delta_{kl} = \left\{\begin{tabular}{cc}
      0, &\it{k $\neq$ l}\\
      1, &\it{k = l}\\
      \end{tabular}\right.
\end{aligned}
\label{cross_corre}
\end{align}
where $\Delta f \overset{\Delta}{=} \frac{1}{T}$ for orthogonal ensurance.

In (\ref{cross_corre}), cross-correlation coefficients for different chirp signals are all zeros, in other words, these chirp waveforms are orthogonal to each other. Hence, we have derived cross-correlation coefficients explicitly in closed-form expressions and proposed chirp waveform design is orthogonal.

\subsection{Orthogonal Chirp Transform}
An orthogonal chirp transform (OCT) is designed on the basis of orthogonal chirp waveforms described as
\begin{align}
\label{OCT}
X(f) =  \sqrt{\frac{1}{T}} \int_{-\infty}^{\infty} x(t)e^{-j(2\pi ft+\pi \mu t^2)}dt,
\end{align}
where $x(t)$ is a continuous-time function and $X(f)$ is its OCT representation in frequency domain. Consequently, OCT can transform a time function into its frequency form.

On the contrary, the inverse orthogonal chirp transform (IOCT) can be written as
\begin{align}
\label{IOCT}
x(t) =  \sqrt{\frac{1}{T}} \int_{-\infty}^{\infty} X(f)e^{j(2\pi ft+\pi \mu t^2)}df.
\end{align}

IOCT can transform a function in frequency into its time-domain representation. Therefore, OCT and IOCT can transform a time-domain form and its frequency representation correspondingly. OCT and IOCT can let our implementations and representations in time domain much more convenient. However, there is still of need for digital or discrete applications. Hence, we should develop their correspondent discrete transforms.

\subsection{Discrete Orthogonal Chirp Transform}
In current implementations, digital forms are more preferable, since it's impractical to store and to process in continuous-time manner, so we develop discrete forms of digital chirp transforms. Therefore, discrete orthogonal chirp transform (DOCT) can be expressed as
\begin{align}
\label{DOCT}
X[k] = \sqrt{\frac{1}{N}} \sum_{n=0}^{N-1} x[n]e^{-j(2\pi kn+\pi \mu n^2)},
\end{align}
where $N$ is the number of OCT points, and $x[n]$ is the $n$-th sample in time. Hence, DOCT can transform digital time signals into frequency forms.

Conversely, inverse discrete orthogonal chirp transform (IDOCT) can be described as
\begin{align}
\label{IDOCT}
x[n] = x(t)|_{t = n T_s}=  \sqrt{\frac{1}{N}} \sum_{k=0}^{N-1} X[k]e^{j(2\pi kn+\pi \mu n^2)},
\end{align}
where $T_s$ is the sampling period. Therefore, DOCT and IDOCT can transform digital signals between time and frequency domains, whereas these two representations compose a DOCT pair.

In practical implementations or signal processing, discrete representations are more demanded, so that's the motivation for our designs for DOCT, which is effective for discrete signal processing in communication systems.

\section{System Model}
\label{S3}

\subsection{MCDM System}
Orthogonal chirp waveforms are utilized in frequency subcarriers in MCDM modulation systems. Hence, the system is benefited by the merits of chirp signals, such as robust to multipath delays and Doppler effect. In addition, it is implemented in multicarrier and multiuser architectures, so MCDM systems can support higher data rates and simultaneous multiuser access. On the other hand, they can be implemented in OCT and DOCT practices.

As a result, passband signals in MU-MCDM systems can be described as
\begin{align}
\label{s_pass}
x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},~0\leq t\leq T
\end{align}
where $E$ is the signal energy, $s[k]$ the $k$-th modulated symbol, $K$ the total number of frequency subcarriers, $\Delta f$ the frequency spacing between subcarriers, $\mu$ the chirp rate, $f_c$ the carrier frequency, and $T$ is the symbol period. $s[k]$ can be a symbol of any modulations, e.g. BPSK, QPSK, and 32-QAM, and its symbol energy is normalized to 1. Hence, signal energy is merely dependent on parameter $E$.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sys_Tx.pdf}
  \caption{MU-MCDM transmitter system model.}
  \label{Sys_Tx}
\end{figure}

Signal representations in (\ref{s_pass}) can be written as the summation of output of IOCT and carrier frequency modulation as in Fig. \ref{Sys_Tx}. The transmitted symbol $s[k]$ is modulated in frequency domain, and after exploiting IOCT, the signals are continuous-time signals, which are prepared to be transmitted in UW-A channels.

Underwater channels are modeled as multipath, independent, and time-variant represented as
\begin{align}
h(t) \overset{\triangle}{=} \sum_{m=1}^{M} h_m(t) \delta(t - \tau_m(t)),
\label{eq:3}
\end{align}
where $M$ is the total number of solvable paths, $h_m(t)$ the $m$-th path's amplitude, and $\tau_m(t)$ is the $m$-th path's delay.

Each path's amplitude and delay are assumed to be time-invariant in a MCDM symbol period. Hence, received signals after carrier demodulation can be described as
\begin{align}
r(t) = \sqrt{E} \sum_{k=0}^{K-1}\sum_{m=1}^{M} {\tilde h}_m s[k]\psi_k(t- \tau_m) + n(t),
\label{eq:4}
\end{align}
where ${\tilde h}_m \overset{\Delta}{=} h_me^{-j2\pi f_c \tau_m} \in \mathbb{C}$ is the $m$-th path's effective channel coefficient, $\tau_m(t)$ the $m$-th path's delay, and $n(t)$ is ambient noise.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Packet.pdf}
  \caption{MU-MCDM transmitted packet structure.}
  \label{packet}
\end{figure}

The transmitted packet structure of a MU-MCDM system is shown in Fig \ref{packet}. There is a preamble block comprising pseudo random antipodal bits $\in \{\pm 1\}^{N_{pn}}$ in the beginning of the transmitted structure for packet synchronization. A MCDM symbol contains multiuser chirp signals in a time duration $T$, and a guard period $T_g$ is inserted between preambles and a MCDM symbol. In addition, zeros are padded after a MCDM symbol with a duration $T_{zp}$, so we consider a ZP-MCDM architecture. The ZP procedure can avoid multipath delays and inter symbol interference (ISI) in UW-A channels.

Our problem objective is to design a coherent receiver with procedures, including synchronization, channel estimation, and symbol detection. Packet synchronization is proceeded by known preambles. Channel state information (CSI) is estimated by pilot symbols among frequency subcarriers. The symbol detection is conducted by estimated channel coefficients and ML symbol detections.

\subsection{Multiuser Access}
Multiuser (MU) access is the practices that enabling multiple users' simultaneous transmission in communication systems. We implement MU access by subcarrier allocation in frequency domain, so multiple users are assigned for different subcarriers and they can transmit their own information symbols in MCDM systems. There are various kinds of MU access scenarios and we focus on block-type and comb-type subcarrier allocation for multiple users.

\subsubsection{Block-type MU Access}
Block-type MU access is conducted by allocating subcarriers to multiple users by groups of subcarrier as depicted in Fig. \ref{sub_block}, e.g. subcarrier 0,1,2 for User 1, subcarrier 3,4,5 for User 2, and subcarrier 6,7,8 for User 3. Inherently, block assignment is convenient for users to manipulate their own transmission in their assigned blocks of subcarriers. However, if channels suffer from notorious selective frequency fading, certain frequency bands may not be able to communicate efficiently. Therefore, this is the motivation for us to exploit alternative MU access scenario.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sub_Block.pdf}
  \caption{Block-type multiuser access.}
  \label{sub_block}
\end{figure}

\subsubsection{Comb-type MU Access}
Subcarriers can also be allocated that each user's subcarriers are distributed uniformly among subcarriers in frequency domain. Hence, in deep fading cases, ineffective frequency bands can distribute among users instead of deteriorating certain user as in block-type MU access. However, the scenario has its drawbacks as well, such as each user demand to manage their information symbols among frequency subcarriers. In this situation, received symbols may suffer from inter symbol interference (ISI) among multiple users.

Therefore, our problem objective is MU-MCDM systems can adapt to choose the parameters, including number of users, number of subcarriers, portion of pilot subcarriers, and MU scenarios in underwater multipath environments. Adaptation can make the system adjust its transmit strategies according to channel's conditions to enhance efficiency in communications. In addition, receiver design comprises packet synchronization, channel estimation, and ML symbol detection procedures.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sub_Comb.pdf}
  \caption{Comb-type multiuser access.}
  \label{sub_comb}
\end{figure}

\section{Receiver Design}
\label{S4}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sys_Rx.pdf}
  \caption{MU-MCDM receiver system model.}
  \label{Sys_Rx}
\end{figure}

Receiver design consists of packet synchronization, channel estimation, and symbol detection illustrated in Fig. \ref{Sys_Rx}. The signal processing is in baseband after carrier frequency demodulation. OCT is conducted to inverse IOCT procedure in the transmitter side. Then, channel estimation and ML detection is processed in frequency.

\subsection{Packet Synchronization}
First, received signals are conducted by carrier frequency demodulation over a time window. Preambles are utilized for packet synchronization, containing $N_{pn}$ antipodal bits in time duration $T_{pn}$. Considered window should at least cover the received preamble symbols for the synchronization purpose.

To synchronize a received packet with a transmitted packet, correlation of transmitted preambles with received signals is written as
\begin{align}
\label{auto}
R_{pn}(\tau) = \int_0^{T_{pn}} x_{pn}(t)r^*(t+\tau)dt,
\end{align}
where $r(t+\tau)$ is the received signal time shifted by $\tau$, $x_{pn}(t)$ is transmitted preamble signal in time, and $\tau \in [0,~T_w]$.

So, the packet is synchronized at the maximum of square L2-norm of the correlated output described as
\begin{align}
\label{k_syn}
{\hat{t}_r} = arg\ \max_{\tau \in [0,~T_w]} ||R_{pn}(\tau)||^2_2.
\end{align}
Consequently, the received packet is synchronization at the time $\hat{t}_r$, so the beginning of a packet's transmission is located.

\subsection{Channel Estimation}
Then, OCT is conducted for analyzing signals in frequency domain. For the $k$-th subcarrier, OCT is conducted in frequency band $\in [k\Delta f,~k\Delta f+B_c]$. Moreover, MCDM symbols are sampled at the time duration $T_s = \frac{1}{f_s}$. The OCT procedure in the receiver is to inverse the IOCT process in the transmitter. The receiver MCDM vector after carrier frequency demodulation can be expressed as
\begin{align}
\label{y_base}
\mathbf{y} = \mathbf{H} \mathbf{s} + \mathbf{n},
\end{align}
where $\mathbf{y} \in \mathbb{C}^{K \times 1}$ is the received vector, $\mathbf{H}\in \mathbb{C}^{K \times K}$ the channel matrix formed as,
\begin{align}
{\mathbf H} \overset{\Delta}{=} \left[\begin{array}{ccccc}
       h_0    & 0      & \cdots  & 0       & 0\\
       0      & h_1    & \cdots  & 0       & 0\\
       \vdots &\vdots  & \ddots  & \vdots  & \vdots\\
       0      & 0      & \cdots  & h_{K-2} & 0\\
       0      & 0      & \cdots  & 0       & h_{K-1}
     \end{array}\right]
\label{eq:6}
\end{align}
$\mathbf{s} \in \mathbb{C}^{K \times 1}$ the transmitted symbol vector, $\mathbf{n}\in \mathbb{C}^{K \times 1}$ the received noise vector, and $K$ is the overall number of frequency subcarriers.

Channel model in a diagonal matrix is built on the assumption that bandwidth of a subcarrier lie within channel's coherence bandwidth \cite{coulson01}, whereas a channel's amplitude of a subcarrier can be redeemed as unchanged in a subcarrier's duration. To be consistent to CSI in our system model, the channel coefficient of a subcarrier should be estimated as a complex number $\in \mathbf{C}$. Therefore, the computational overhead of following channel estimation and symbol detection decreases dramatically in MCDM systems.

Pilot symbols are arranged in comb-type subcarrier allocation \cite{shen06}, which pilots are spread evenly among subcarriers and subcarriers' allocation does not change over time. Total number of pilot symbols $K_p$, they are located at $k$ = $0$, $L$, $2L$, ~\ldots~,$(K_p-1)L$, where $L \overset{\Delta}{=} \frac{K}{K_p}$. Therefore, channel states are estimated every $L$ subcarrier for capturing CSI effectively.

Then, the received pilot vector in baseband can be described as
\begin{align}
\label{y_pilot}
\mathbf{y}_{\it p} = \mathbf{S}_{\it p} \mathbf{h}_{\it p} + \mathbf{n}_{\it p},
\end{align}
where $\mathbf{y}_p \in \mathbb{C}^{K_p \times 1}$ is the received pilot vector, $\mathbf{h}_{\it p} = [h_{p,0},\ h_{p,1},\ \cdots,\ h_{p,K_p-1}]^T \in \mathbb{C}^{K_p \times 1}$ the pilot channel vector, $\mathbf{S}_{\it p}\in \mathbb{C}^{K_p \times K_p}$ the pilot symbols in a diagonal matrix form,
\begin{align}
{\mathbf S_p} \overset{\Delta}{=} \left[\begin{array}{ccccc}
       s_{p,0} &0       &\cdots  &0\\
       0       &s_{p,1}&\cdots   &0\\
       \vdots  &\vdots  &\ddots  &\vdots\\
       0       &0       &\cdots  &s_{p,K_p-1}
     \end{array}\right]
\label{s_p}
\end{align}
and $\mathbf{n}_{\it p} \in \mathbb{C}^{K_p \times 1}$ is the pilot noise vector.

Moreover, channel estimation for pilot symbols can be written as
\begin{align}
\label{pro_che}
\mathbf{\widehat{h}_{\it p}} &= arg\ \min_{\mathbf{h_p} \in \mathbb{C}^{K_p \times 1}} ||\mathbf{y}_{\it p} - \mathbf{S}_{\it p} \mathbf{h}_{\it p}||^2_2.
\end{align}

Pilot matrix $\mathbf{S}_{\it p}$ is known a-priori. So under additive white Gaussian noise (AWGN) condition, ML-optimal channel estimator \cite{Proakis07} can be expressed as
\begin{align}
\label{h_ML}
\mathbf{\widehat{h}_{\it p,ML}} &= (\mathbf{S}_p^H \mathbf{S}_p)^{-1}\mathbf{S}_p^H \mathbf{y}_p.
\end{align}

ML channel vector for pilot symbols is estimated and we make assumption that a subcarrier's channel is narrowband and channel coefficients vary linearly between neighbored subcarriers. Hence, linear interpolation is conducted for channel coefficients for the non-pilot subcarrier $k$ as
\begin{align}
\label{h_inter}
\widehat{h}_{ML}[k] = \left(1-\frac{l}{L}\right)\widehat{h}_{p,ML}[m] + \frac{l}{L}\widehat{h}_{p,ML}[m+1],
\end{align}
where $\widehat{h}_{p,ML}[m]$ is the estimated channel coefficient of the $m$-th pilot symbol, $l$ the residue of $k$ divided by $L$, $mL < k < (m+1)L$, and $m = 0,\cdots, K_p-1$.

Consequently, channel coefficients of all the frequency subcarriers are estimated, so symbol detection can be further processed.

\subsection{ML Symbol Detection}
Our detection objective is to minimize the error between received vector $\mathbf{y}$ and channel-filtered vector $\mathbf{Hs}$ described as
\begin{align}
\label{prob}
\mathbf{\widehat{s}} &= arg\ \min_{\mathbf{s} \in \mathbb{C}^{K \times 1}} ||\mathbf{y} - \mathbf{H} \mathbf{s}||^2_2.
\end{align}

In AWGN, the ML-optimal detection for (\ref{prob}) can be written in a closed-form expression as
\begin{align}
\label{s_ML}
\mathbf{\widehat{s}_{\it ML}} &= (\mathbf{H}^H \mathbf{H})^{-1} \mathbf{H}^H \mathbf{y},
\end{align}
where $\mathbf{H}$ is estimated by channel coefficients in (\ref{h_ML}) and (\ref{h_inter}), and then formulated in a diagonal matrix as (\ref{eq:6}).

As a result, a low complexity receiver design is proposed, including packet synchronization, OCT process, channel estimation, and ML symbol detection. Then, MU-MCDM systems' performance demands to be evaluated.

\section{Discussion}
\label{S5}
Several implementations for our proposed approaches are discussed and analyzed. OCT can enable continuous-time practices, symbol transmission, and channel model in time-domain. On the other hand, DOCT can let us to proceed digital practices and signal processing in discrete manner. More details are described in th following.

\subsection{Relationships of continuous and discrete OCT}
Traditionally, OCT is defined as implementations and applications utilized in continuous-time and system design. Moreover, for transmitter system model or antenna design, they are all time-domain procedures, so we can handle them with OCT and other continuous-time operations. On the other hand, various practical implementations are in need in discrete or digital forms, e.g., digital signal processing, so we develop DOCT for the demand of correspondent digital applications.

\subsection{Compatibility to OFDM Architectures}
MCDM systems are compatible with prevailing OFDM architectures, so this let our approaches as promising and alternative scenarios for applying in current dominant multicarrier systems. We only need to include the frequency sweeping term, i.e., chirped signals, for each frequency subcarrier as illustrated in Fig..

\section{Simulation Studies}
\label{S6}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Ch_Sim.pdf}
  \caption{Normalized channel response in simulations.}
  \label{ch_Sim}
\end{figure}

In simulations, MCDM systems are simulated in UW-A multipath emulators. Channel's amplitude is modeled as Rayleigh distribution; noise is AWGN, and 4 resolvable paths are shown in Fig. \ref{ch_Sim}. We consider several modulations, including binary phase shift keying (BPSK) and quadrature PSK (QPSK). Moreover, bit mapping for symbols is gray coded for reducing detection errors in adjacent symbols. Total number of subcarriers $K$ can be adaptive to choose from 128, 256, 512, and 1024; portions of pilot subcarriers are available for $\frac{1}{32}$, $\frac{1}{16}$, $\frac{1}{8}$, and $\frac{1}{4}$. Carrier frequency is $f_c$ = 100 kHz. The frequency spacing between subcarriers $\Delta f$ can be 64, 128, 255, 509 Hz. The MCDM symbol period $T$ can be 1.97, 3.93, 7.86, and 15.73 ms, the preamble duration $T_{pn}$ = 1.31 ms, the pause interval between preambles and a MCDM symbol $T_p$ = 1.54 ms, and the guard interval between MCDM symbols $T_g$ = 2.56 ms. Linear up-chirps are applied, chirp rate $\mu$ can be $8.94 \times 10^3$, $1.79 \times 10^4$, $3.58 \times 10^4$, $7.15 \times 10^4$ Hz/s. Available bandwidth for the MCDM system is 65.25 kHz. The simulation results are averaged over 1000 recorded packets.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_K.pdf}
  \caption{MCDM for different number of subcarriers.}
  \label{Sim_K}
\end{figure}

Fig. \ref{Sim_K} shows BER vs. SNRs for different number of subcarriers $K$. The portion of subcarriers is fixed at $\frac{1}{4}$ for different $K$. BER indicated here is the average BER for all users (URs). If we compare $K$ = 1024 with $K$ = 512, BER is decreased for 1.58 dB at SNR = 12 dB. The more the number of subcarrier, the lower the BER in MCDM systems. Among the available number of subcarriers, BER can be reach to 1.28 $\times 10^{-2}$ at $K$ = 1024 at SNR = 12 dB.

\begin{figure}
 \centering
  \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_K_256_Kp.pdf}
  \caption{K = 256 for different number of pilots.}
  \label{Sim_256_Kp}
\end{figure}

In Fig. \ref{Sim_256_Kp} illustrates BER to SNR values of different pilots' portion for $K$ = 256. BER improvement for $K_p$ = 16 and $K_p$ = 8 is 0.45 dB, while that for $K_p$ = 64 and $K_p$ = 32 is 1.55 dB. BER decrease between adjacent curves is larger for higher $K_p$. For fixed $K$, as the number of pilots increases, respective BER is reduced significantly since CSI can estimated more accurately by increasing number of pilot symbols.

\begin{figure}
 \centering
  \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_K_512_Kp.pdf}
  \caption{K = 512 for different number of pilots.}
  \label{Sim_512_Kp}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_K_1024_Kp.pdf}
  \caption{K = 1024 for different number of pilots.}
  \label{Sim_1024_Kp}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_PSK_K.pdf}
  \caption{PSK-MCDM for different number of subcarriers.}
  \label{Sim_PSK_K}
\end{figure}

Fig. \ref{Sim_512_Kp} shows BER of different pilots' portion for $K$ = 512. BER of $K$ = 128 is improved for 1.95 dB to that of $K$ = 64 at SNR = 10 dB, whereas BER improvement for $K$ = 64 and $K$ = 128 is 1.52 dB at SNR = 12 dB. In the $K$ = 512 case, BER follows the trend as well that BER is reduced for larger number of pilots.

Fig. \ref{Sim_1024_Kp} demonstrates BER of different pilots' portion for $K$ = 1024. As the same as previous simulation results, higher number of pilot symbols can benefit MCDM systems in BER performance. Moreover, at SNR = 10 dB, BER reduction of $K_p$ = 128 to that of $K_p$ = 256 is 0.83 dB, which is smaller than that of $K_p$ = 64 to BER of $K_p$ = 128 2.68 dB. I suppose the rationale is that average CSI variants among subcarriers is around 4 to 8 subcarrier interval. Hence, $K_p$ = 256 can update channel information every 4 subcarriers, which some updates of channel coefficients cannot benefit MCDM systems as expected in detections.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Block_UR_3.pdf}
  \caption{Block-type access with $N_{UR}$ = 3.}
  \label{Sim_block_3}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Block_UR_4.pdf}
  \caption{Block-type access with $N_{UR}$ = 4.}
  \label{Sim_block_4}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Block_Adaptive.pdf}
  \caption{Adaptive MCDM in block-type access.}
  \label{Sim_block_adap}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Comb_UR_3.pdf}
  \caption{Comb-type access with $N_{UR}$ = 3.}
  \label{Sim_comb_3}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Comb_UR_4.pdf}
  \caption{Comb-type access with $N_{UR}$ = 4.}
  \label{Sim_comb_4}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Comb_Adaptive.pdf}
  \caption{Adaptive MCDM in comb-type access.}
  \label{Sim_comb_adap}
\end{figure}

Fig. \ref{Sim_PSK_K} demonstrates BER of PSK-MCDM for $K$ = 256 and 512. If we compare BPSK, $K$ = 512 can decrease BER for 1.18 dB to that of $K$ = 256 at SNR = 10 dB. Also, for $K$ = 512, QPSK can increase BER for 3.30 dB than BPSK at SNR = 12 dB, while bit rate of QPSK can be two times to that of BPSK. Hence, it is a trade-off for BER and transmission rate in designing MCDM systems. All in all, higher number of subcarrriers can enhance BER performance in MCDM systems.

Fig. \ref{Sim_block_3} illustrates BER vs. SNR for $N_{UR}$ = 3 for $K$ = 256 in block-type MU access. Subcarriers are allocated as first one-third symbol carriers to UR 1, second one third to UR 2, and third one third to UR 3. The subcarrier allocation is fixed for 1000 packets' transmission. As we can observe from the figure, UR 1 and UR 2 perform worse in BER, while UR 3 can achieve lower BER values. This can be explained as in the multipath Rayleigh fading channel's realizations, UR 3's block of subcarriers suffer less from frequency fading, so it can reach lower BER values in detections.

Fig. \ref{Sim_block_4} shows BER vs. SNR values for $N_{UR}$ = 4 for $K$ = 256 in block-type MU access. Here, subcarriers are divided into four blocks and assigned to UR 1, 2, 3, and 4, respectively. As we can observe from the figure, UR 1, 3, and 4 can achieve lower BER, while UR 2 obtain worse BER. BER of UR 3 is improved for 3.19 dB to that of UR 2 at SNR = 12 dB.

Fig. \ref{Sim_block_adap} demonstrates average BER for adaptive MCDM for available $K$ = 128, 256, 512, and 1024 in block-type MU access, SNR fixed at 12 dB. In the practice, there are 10 packets transmitted in multipath fading channels. Packet index 1 to 4, the MCDM system utilize $K$ = 128 for transmission; packet index 5 and 6 for $K$ = 256; index 7 and 8 for $K$ = 512; index 9 and 10 for $K$ = 1024. Based on the simulation results, the MCDM system decides to adopt $K$ = 1024 for future transmission. The feedback information is transmitted to the transmitter through a low-rate and robust chirp signals. Hence, the transmitter can adapt to the scenario enhancing BER performance most.

Fig. \ref{Sim_comb_3} shows BER for $N_{UR}$ = 3 for $K$ = 256 in comb-type MU access. Users' subcarriers are assigned in interleaving manner, e.g., if we have 6 available subcarriers with indexes 0 - 5, subcarrier 0 and 3 for UR 1, subcarrier 1 and 4 for UR 2, subcarrier 2 and 5 for UR 3. Then, the subcarrier allocation is fixed during all the transmission. UR 2 and UR 3 have higher BER values, whereas UR 1's BER is the lowest among the users. During the transmission, UR 1's subcarriers suffer less from channel fading, so its BER values are the best in the simulations.

Fig. \ref{Sim_comb_4} illustrates BER for $N_{UR}$ = 4 for $K$ = 256 in comb-type MU access. Subcarriers are assigned to multiple users by the interleaving procedure. In the practice, UR 2 can decrease BER by 3.11 dB to that of UR 3 at SNR = 8 dB, while BER improvement for UR 1 and UR 4 is merely 0.58 dB. The simulation results are influenced by fading channel characteristics on frequency subcarriers in frequency. If we compare the lowest BER among users in comb-type with block-type MU access, comb-type can improved for 2.71 dB. This can be explained that in flat-fading channels, since comb-type allocation distributes the subcarriers uniformly among users, all the users can be beneficial to their performance.

Fig. \ref{Sim_comb_adap} demonstrates average BER for adaptive MCDM for available $K$ = 128, 256, 512, and 1024 in comb-type MU access, SNR fixed at 12 dB. There are 10 packets transmitted in the simulation. Packet index 1 to 4, the MCDM system utilize $K$ = 128 for transmission; packet index 5 and 6 for $K$ = 256; index 7 and 8 for $K$ = 512; ; index 9 and 10 for $K$ = 1024. Based on the simulation results, the MCDM system decides to adopt $K$ = 1024 for future transmission. Comb-type can decrease BER by 0.24 dB than that of block-type.

\section{Conclusion}
\label{S7}
In this work, we propose to apply orthogonal chirp waveforms as subcarriers in MCDM systems. Additionally, the proposed communication systems are able to support multiple users' transmission simultaneously. Adaptive transmission strategies are as well considered during the data transmission for optimizing BER performance, including number of subcarriers, number of pilot subcarriers, digital modulations, number of users, approaches to deploy pilot subcarriers. Simulation studies demonstrate that adaptive transmission scenarios outperform other scenarios. The BER performance of the adaptive MU-MCDM is the best among various scenarios. In conclusion, adaptive transmission strategies optimize the performance for MU-MCDM systems.

\bibliographystyle{IEEEtran}

\section{Introduction}
\label{S1}

Underwater acoustic (UW-A) communication has been developed for various applications, including offshore remote control, undersea pollution control, environmental monitoring, underwater communication and sensor networks,  surveillance and defense \cite{melodia13,demirors15,akyildiz04,stojanovic96}.

Currently, there are still many challenges in UW-A channels, such as severe attenuation loss, multipath propagation delay, ambient noise, Doppler effect, and frequency selective fading that restrict our technology from advancing significantly \cite{melodia13}.

Chirp signals have been utilized widely in radar and sonar applications \cite{fitzgerald74,lurton02} primarily due to their resilience to multipath propagation delays and Doppler spread as well as superior correlation characteristics. Applications in UW-A communications include linear chirp signals for navigation and underwater probing \cite{austin94,chiu15}, packet synchronization \cite{sharif00,kilfoyle00}, channel estimation \cite{wu12}, reliable UW-A communications \cite{he09}, Doppler estimation and compensation \cite{lurton02,sharif00}, and robust feedback communications \cite{demirors14}.

However, linear chirps are not orthogonal rigorously \cite{springer00}. To tackle this problem, some researchers adopt various categories of orthogonal chirp transforms, such as fractional Fourier transform \cite{en14,solyman12}, and Fresnel transform \cite{ouyang16} for supporting multicarrier communications and high data rates. However, the disadvantages of these chirp transforms are high weighted computation and complex implementations. Hence, there is of interests to design orthogonal chirp waveforms and low complexity receivers in UW-A environments \cite{huang17}.

Propelled by the need for attaining higher data rates and multiple users, chirp signals have been utilized in multicarrier communications. Some researchers applied chirps in OFDM for enhancing Doppler spread estimation, and delay tolerance \cite{dida16,barbarossa01}. Some focus on waveform designs for multiple transmitters in radar systems \cite{kim13}, orthogonal chirp signals in multiple input multiple output (MIMO) systems \cite{wang15}, and CSS for enhancing orthogonality of transmission of different users \cite{yang11,cheng15}. Several works are built on different categories of chirp bases in multicarrier systems, e.g., fractional Fourier transform (FrFT), fractional cosine transform, and Fresnel chirp transform \cite{en04,solyman12,attar17,ouyang16}. On the other hand, chirps can be exploited in multiuser communications for multiple access \cite{shen06_div}. Multicode scheme can enable simultaneous transmission of multiple chirp waveforms \cite{liu06}. However, these chirp transforms are of high computational overhead and complex implementations. Hence, developing an efficient and low complexity multicarrier chirp system is still of interests and researches in RF wireless communications.

In this paper, we consider a single-input single-output (SISO) UW-A communications that information symbols are carried over orthogonal chirp frequency subcarriers in adaptive multiuser multicarrier chirp-division multiplexing (MU-MCDM) systems. The orthogonality of chirp waveforms are analyzed by their cross-correlation coefficients. Both continuous orthogonal chirp transform (OCT) and discrete orthogonal chirp transform (DOCT) are introduced for time-domain or digital implementations. The system is adaptive in preamble duration, number of subcarriers, portion of pilots, guard interval, and multiuser scenario to optimize performance of transmission in variant underwater environments. Moreover, we develop a low complexity coherent receiver of channel estimation and maximum-likelihood (ML) symbol detection. On the other hand, we discuss the relationships of OCT and DOCT and their various applications. In addition, MCDM system is compatible with prevailing OFDM architectures. Then, simulation studies are demonstrated over UW-A multipath channels for evaluating adaptive MU-MCDM systems.

\vspace{2pt}
The rest of the paper is organized as follows. Section \ref{S2} addresses OCT representations. System Model is described in Section. \ref{S3}. Receiver design of the ML coherent receiver is introduced in Section \ref{S4}. Discussion of proposed approaches are illustrated in Section \ref{S5}. Simulation results are demonstrated in Section \ref{S6}. Finally, several concluding remarks are drawn in Section \ref{S7}.

\section{Orthogonal Chirp Waveform Design}
\label{S2}
\subsection{Chirp Waveform Design}
A linear chirp signal is considered, where waveform's frequency change linearly over time. The $m$-th chirp waveform can be represented as
\begin{align}
\psi_m(t) =  \sqrt{\frac{1}{T}} e^{j(2\pi m\Delta ft+\pi \mu t^2)},~0\leq t\leq T
\label{psi}
\end{align}
where $T$ is the symbol duration, $\Delta f$ the frequency spacing between chirps, $\mu\overset{\Delta}{=}\frac{B_c}{T}$ the chirp rate, $B_c$ the bandwidth of a chirp signal, and the waveform's energy is normalized to 1. $m\Delta f$ is the initial frequency of the $m$-th chirp waveform. For the chirp rate, $\mu >$ 0 is a up-chirp signal (frequency growing with time), while $\mu <$ 0 is a down-chirp signal (frequency reducing with time).

Proposed chirp signals are complex, comprising both in-phase and quadrature-phase parts, so as we process channel estimation, channel coefficients should be treated as complex representations. Moreover, additional bandwidth assigned to chirp signals can support a degree of freedom for resilience of multipath effect and channel fading. Then, we examine the orthogonality of chirp waveforms in derivations.

\subsection{Orthogonality of Chirp Waveforms}
Characteristics of orthogonality of chirp waveforms are crucial in our waveform designs, so we analyze the orthogonality of chirp waveforms with cross-correlation coefficients. Then, the cross-correlation coefficient of the $k$-th and the $l$-th chirp waveforms can be derived as
\begin{align}
\begin{aligned}
\rho_{kl}  &= \int_0^{T} \psi_{k}(t) \psi_{l}^*(t)dt\\

      &= \frac{1}{T} \int_0^{T} e^{j2\pi (k-l)\Delta ft}dt\\
      &= \delta_{kl} = \left\{\begin{tabular}{cc}
      0, &\it{k $\neq$ l}\\
      1, &\it{k = l}\\
      \end{tabular}\right.
\end{aligned}
\label{cross_corre}
\end{align}
where $\Delta f \overset{\Delta}{=} \frac{1}{T}$ for orthogonal ensurance.

In (\ref{cross_corre}), cross-correlation coefficients for different chirp signals are all zeros, in other words, these chirp waveforms are orthogonal to each other. Hence, we have derived cross-correlation coefficients explicitly in closed-form expressions and proposed chirp waveform design is orthogonal.

\subsection{Orthogonal Chirp Transform}
An orthogonal chirp transform (OCT) is designed on the basis of orthogonal chirp waveforms described as
\begin{align}
\label{OCT}
X(f) =  \sqrt{\frac{1}{T}} \int_{-\infty}^{\infty} x(t)e^{-j(2\pi ft+\pi \mu t^2)}dt,
\end{align}
where $x(t)$ is a continuous-time function and $X(f)$ is its OCT representation in frequency domain. Consequently, OCT can transform a time function into its frequency form.

On the contrary, the inverse orthogonal chirp transform (IOCT) can be written as
\begin{align}
\label{IOCT}
x(t) =  \sqrt{\frac{1}{T}} \int_{-\infty}^{\infty} X(f)e^{j(2\pi ft+\pi \mu t^2)}df.
\end{align}

IOCT can transform a function in frequency into its time-domain representation. Therefore, OCT and IOCT can transform a time-domain form and its frequency representation correspondingly. OCT and IOCT can let our implementations and representations in time domain much more convenient. However, there is still of need for digital or discrete applications. Hence, we should develop their correspondent discrete transforms.

\subsection{Discrete Orthogonal Chirp Transform}
In current implementations, digital forms are more preferable, since it's impractical to store and to process in continuous-time manner, so we develop discrete forms of digital chirp transforms. Therefore, discrete orthogonal chirp transform (DOCT) can be expressed as
\begin{align}
\label{DOCT}
X[k] = \sqrt{\frac{1}{N}} \sum_{n=0}^{N-1} x[n]e^{-j(2\pi kn+\pi \mu n^2)},
\end{align}
where $N$ is the number of OCT points, and $x[n]$ is the $n$-th sample in time. Hence, DOCT can transform digital time signals into frequency forms.

Conversely, inverse discrete orthogonal chirp transform (IDOCT) can be described as
\begin{align}
\label{IDOCT}
x[n] = x(t)|_{t = n T_s}=  \sqrt{\frac{1}{N}} \sum_{k=0}^{N-1} X[k]e^{j(2\pi kn+\pi \mu n^2)},
\end{align}
where $T_s$ is the sampling period. Therefore, DOCT and IDOCT can transform digital signals between time and frequency domains, whereas these two representations compose a DOCT pair.

In practical implementations or signal processing, discrete representations are more demanded, so that's the motivation for our designs for DOCT, which is effective for discrete signal processing in communication systems.

\section{System Model}
\label{S3}

\subsection{MCDM System}
Orthogonal chirp waveforms are utilized in frequency subcarriers in MCDM modulation systems. Hence, the system is benefited by the merits of chirp signals, such as robust to multipath delays and Doppler effect. In addition, it is implemented in multicarrier and multiuser architectures, so MCDM systems can support higher data rates and simultaneous multiuser access. On the other hand, they can be implemented in OCT and DOCT practices.

As a result, passband signals in MU-MCDM systems can be described as
\begin{align}
\label{s_pass}
x(t) = \sqrt{E}\sum_{k=0}^{K-1} s[k]\psi_k(t) e^{j2\pi f_c t},~0\leq t\leq T
\end{align}
where $E$ is the signal energy, $s[k]$ the $k$-th modulated symbol, $K$ the total number of frequency subcarriers, $\Delta f$ the frequency spacing between subcarriers, $\mu$ the chirp rate, $f_c$ the carrier frequency, and $T$ is the symbol period. $s[k]$ can be a symbol of any modulations, e.g. BPSK, QPSK, and 32-QAM, and its symbol energy is normalized to 1. Hence, signal energy is merely dependent on parameter $E$.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sys_Tx.pdf}
  \caption{MU-MCDM transmitter system model.}
  \label{Sys_Tx}
\end{figure}

Signal representations in (\ref{s_pass}) can be written as the summation of output of IOCT and carrier frequency modulation as in Fig. \ref{Sys_Tx}. The transmitted symbol $s[k]$ is modulated in frequency domain, and after exploiting IOCT, the signals are continuous-time signals, which are prepared to be transmitted in UW-A channels.

Underwater channels are modeled as multipath, independent, and time-variant represented as
\begin{align}
h(t) \overset{\triangle}{=} \sum_{m=1}^{M} h_m(t) \delta(t - \tau_m(t)),
\label{eq:3}
\end{align}
where $M$ is the total number of solvable paths, $h_m(t)$ the $m$-th path's amplitude, and $\tau_m(t)$ is the $m$-th path's delay.

Each path's amplitude and delay are assumed to be time-invariant in a MCDM symbol period. Hence, received signals after carrier demodulation can be described as
\begin{align}
r(t) = \sqrt{E} \sum_{k=0}^{K-1}\sum_{m=1}^{M} {\tilde h}_m s[k]\psi_k(t- \tau_m) + n(t),
\label{eq:4}
\end{align}
where ${\tilde h}_m \overset{\Delta}{=} h_me^{-j2\pi f_c \tau_m} \in \mathbb{C}$ is the $m$-th path's effective channel coefficient, $\tau_m(t)$ the $m$-th path's delay, and $n(t)$ is ambient noise.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Packet.pdf}
  \caption{MU-MCDM transmitted packet structure.}
  \label{packet}
\end{figure}

The transmitted packet structure of a MU-MCDM system is shown in Fig \ref{packet}. There is a preamble block comprising pseudo random antipodal bits $\in \{\pm 1\}^{N_{pn}}$ in the beginning of the transmitted structure for packet synchronization. A MCDM symbol contains multiuser chirp signals in a time duration $T$, and a guard period $T_g$ is inserted between preambles and a MCDM symbol. In addition, zeros are padded after a MCDM symbol with a duration $T_{zp}$, so we consider a ZP-MCDM architecture. The ZP procedure can avoid multipath delays and inter symbol interference (ISI) in UW-A channels.

Our problem objective is to design a coherent receiver with procedures, including synchronization, channel estimation, and symbol detection. Packet synchronization is proceeded by known preambles. Channel state information (CSI) is estimated by pilot symbols among frequency subcarriers. The symbol detection is conducted by estimated channel coefficients and ML symbol detections.

\subsection{Multiuser Access}
Multiuser (MU) access is the practices that enabling multiple users' simultaneous transmission in communication systems. We implement MU access by subcarrier allocation in frequency domain, so multiple users are assigned for different subcarriers and they can transmit their own information symbols in MCDM systems. There are various kinds of MU access scenarios and we focus on block-type and comb-type subcarrier allocation for multiple users.

\subsubsection{Block-type MU Access}
Block-type MU access is conducted by allocating subcarriers to multiple users by groups of subcarrier as depicted in Fig. \ref{sub_block}, e.g. subcarrier 0,1,2 for User 1, subcarrier 3,4,5 for User 2, and subcarrier 6,7,8 for User 3. Inherently, block assignment is convenient for users to manipulate their own transmission in their assigned blocks of subcarriers. However, if channels suffer from notorious selective frequency fading, certain frequency bands may not be able to communicate efficiently. Therefore, this is the motivation for us to exploit alternative MU access scenario.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sub_Block.pdf}
  \caption{Block-type multiuser access.}
  \label{sub_block}
\end{figure}

\subsubsection{Comb-type MU Access}
Subcarriers can also be allocated that each user's subcarriers are distributed uniformly among subcarriers in frequency domain. Hence, in deep fading cases, ineffective frequency bands can distribute among users instead of deteriorating certain user as in block-type MU access. However, the scenario has its drawbacks as well, such as each user demand to manage their information symbols among frequency subcarriers. In this situation, received symbols may suffer from inter symbol interference (ISI) among multiple users.

Therefore, our problem objective is MU-MCDM systems can adapt to choose the parameters, including number of users, number of subcarriers, portion of pilot subcarriers, and MU scenarios in underwater multipath environments. Adaptation can make the system adjust its transmit strategies according to channel's conditions to enhance efficiency in communications. In addition, receiver design comprises packet synchronization, channel estimation, and ML symbol detection procedures.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sub_Comb.pdf}
  \caption{Comb-type multiuser access.}
  \label{sub_comb}
\end{figure}

\section{Receiver Design}
\label{S4}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sys_Rx.pdf}
  \caption{MU-MCDM receiver system model.}
  \label{Sys_Rx}
\end{figure}

Receiver design consists of packet synchronization, channel estimation, and symbol detection illustrated in Fig. \ref{Sys_Rx}. The signal processing is in baseband after carrier frequency demodulation. OCT is conducted to inverse IOCT procedure in the transmitter side. Then, channel estimation and ML detection is processed in frequency.

\subsection{Packet Synchronization}
First, received signals are conducted by carrier frequency demodulation over a time window. Preambles are utilized for packet synchronization, containing $N_{pn}$ antipodal bits in time duration $T_{pn}$. Considered window should at least cover the received preamble symbols for the synchronization purpose.

To synchronize a received packet with a transmitted packet, correlation of transmitted preambles with received signals is written as
\begin{align}
\label{auto}
R_{pn}(\tau) = \int_0^{T_{pn}} x_{pn}(t)r^*(t+\tau)dt,
\end{align}
where $r(t+\tau)$ is the received signal time shifted by $\tau$, $x_{pn}(t)$ is transmitted preamble signal in time, and $\tau \in [0,~T_w]$.

So, the packet is synchronized at the maximum of square L2-norm of the correlated output described as
\begin{align}
\label{k_syn}
{\hat{t}_r} = arg\ \max_{\tau \in [0,~T_w]} ||R_{pn}(\tau)||^2_2.
\end{align}
Consequently, the received packet is synchronization at the time $\hat{t}_r$, so the beginning of a packet's transmission is located.

\subsection{Channel Estimation}
Then, OCT is conducted for analyzing signals in frequency domain. For the $k$-th subcarrier, OCT is conducted in frequency band $\in [k\Delta f,~k\Delta f+B_c]$. Moreover, MCDM symbols are sampled at the time duration $T_s = \frac{1}{f_s}$. The OCT procedure in the receiver is to inverse the IOCT process in the transmitter. The receiver MCDM vector after carrier frequency demodulation can be expressed as
\begin{align}
\label{y_base}
\mathbf{y} = \mathbf{H} \mathbf{s} + \mathbf{n},
\end{align}
where $\mathbf{y} \in \mathbb{C}^{K \times 1}$ is the received vector, $\mathbf{H}\in \mathbb{C}^{K \times K}$ the channel matrix formed as,
\begin{align}
{\mathbf H} \overset{\Delta}{=} \left[\begin{array}{ccccc}
       h_0    & 0      & \cdots  & 0       & 0\\
       0      & h_1    & \cdots  & 0       & 0\\
       \vdots &\vdots  & \ddots  & \vdots  & \vdots\\
       0      & 0      & \cdots  & h_{K-2} & 0\\
       0      & 0      & \cdots  & 0       & h_{K-1}
     \end{array}\right]
\label{eq:6}
\end{align}
$\mathbf{s} \in \mathbb{C}^{K \times 1}$ the transmitted symbol vector, $\mathbf{n}\in \mathbb{C}^{K \times 1}$ the received noise vector, and $K$ is the overall number of frequency subcarriers.

Channel model in a diagonal matrix is built on the assumption that bandwidth of a subcarrier lie within channel's coherence bandwidth \cite{coulson01}, whereas a channel's amplitude of a subcarrier can be redeemed as unchanged in a subcarrier's duration. To be consistent to CSI in our system model, the channel coefficient of a subcarrier should be estimated as a complex number $\in \mathbf{C}$. Therefore, the computational overhead of following channel estimation and symbol detection decreases dramatically in MCDM systems.

Pilot symbols are arranged in comb-type subcarrier allocation \cite{shen06}, which pilots are spread evenly among subcarriers and subcarriers' allocation does not change over time. Total number of pilot symbols $K_p$, they are located at $k$ = $0$, $L$, $2L$, ~\ldots~,$(K_p-1)L$, where $L \overset{\Delta}{=} \frac{K}{K_p}$. Therefore, channel states are estimated every $L$ subcarrier for capturing CSI effectively.

Then, the received pilot vector in baseband can be described as
\begin{align}
\label{y_pilot}
\mathbf{y}_{\it p} = \mathbf{S}_{\it p} \mathbf{h}_{\it p} + \mathbf{n}_{\it p},
\end{align}
where $\mathbf{y}_p \in \mathbb{C}^{K_p \times 1}$ is the received pilot vector, $\mathbf{h}_{\it p} = [h_{p,0},\ h_{p,1},\ \cdots,\ h_{p,K_p-1}]^T \in \mathbb{C}^{K_p \times 1}$ the pilot channel vector, $\mathbf{S}_{\it p}\in \mathbb{C}^{K_p \times K_p}$ the pilot symbols in a diagonal matrix form,
\begin{align}
{\mathbf S_p} \overset{\Delta}{=} \left[\begin{array}{ccccc}
       s_{p,0} &0       &\cdots  &0\\
       0       &s_{p,1}&\cdots   &0\\
       \vdots  &\vdots  &\ddots  &\vdots\\
       0       &0       &\cdots  &s_{p,K_p-1}
     \end{array}\right]
\label{s_p}
\end{align}
and $\mathbf{n}_{\it p} \in \mathbb{C}^{K_p \times 1}$ is the pilot noise vector.

Moreover, channel estimation for pilot symbols can be written as
\begin{align}
\label{pro_che}
\mathbf{\widehat{h}_{\it p}} &= arg\ \min_{\mathbf{h_p} \in \mathbb{C}^{K_p \times 1}} ||\mathbf{y}_{\it p} - \mathbf{S}_{\it p} \mathbf{h}_{\it p}||^2_2.
\end{align}

Pilot matrix $\mathbf{S}_{\it p}$ is known a-priori. So under additive white Gaussian noise (AWGN) condition, ML-optimal channel estimator \cite{Proakis07} can be expressed as
\begin{align}
\label{h_ML}
\mathbf{\widehat{h}_{\it p,ML}} &= (\mathbf{S}_p^H \mathbf{S}_p)^{-1}\mathbf{S}_p^H \mathbf{y}_p.
\end{align}

ML channel vector for pilot symbols is estimated and we make assumption that a subcarrier's channel is narrowband and channel coefficients vary linearly between neighbored subcarriers. Hence, linear interpolation is conducted for channel coefficients for the non-pilot subcarrier $k$ as
\begin{align}
\label{h_inter}
\widehat{h}_{ML}[k] = \left(1-\frac{l}{L}\right)\widehat{h}_{p,ML}[m] + \frac{l}{L}\widehat{h}_{p,ML}[m+1],
\end{align}
where $\widehat{h}_{p,ML}[m]$ is the estimated channel coefficient of the $m$-th pilot symbol, $l$ the residue of $k$ divided by $L$, $mL < k < (m+1)L$, and $m = 0,\cdots, K_p-1$.

Consequently, channel coefficients of all the frequency subcarriers are estimated, so symbol detection can be further processed.

\subsection{ML Symbol Detection}
Our detection objective is to minimize the error between received vector $\mathbf{y}$ and channel-filtered vector $\mathbf{Hs}$ described as
\begin{align}
\label{prob}
\mathbf{\widehat{s}} &= arg\ \min_{\mathbf{s} \in \mathbb{C}^{K \times 1}} ||\mathbf{y} - \mathbf{H} \mathbf{s}||^2_2.
\end{align}

In AWGN, the ML-optimal detection for (\ref{prob}) can be written in a closed-form expression as
\begin{align}
\label{s_ML}
\mathbf{\widehat{s}_{\it ML}} &= (\mathbf{H}^H \mathbf{H})^{-1} \mathbf{H}^H \mathbf{y},
\end{align}
where $\mathbf{H}$ is estimated by channel coefficients in (\ref{h_ML}) and (\ref{h_inter}), and then formulated in a diagonal matrix as (\ref{eq:6}).

As a result, a low complexity receiver design is proposed, including packet synchronization, OCT process, channel estimation, and ML symbol detection. Then, MU-MCDM systems' performance demands to be evaluated.

\section{Discussion}
\label{S5}
Several implementations for our proposed approaches are discussed and analyzed. OCT can enable continuous-time practices, symbol transmission, and channel model in time-domain. On the other hand, DOCT can let us to proceed digital practices and signal processing in discrete manner. More details are described in th following.

\subsection{Relationships of continuous and discrete OCT}
Traditionally, OCT is defined as implementations and applications utilized in continuous-time and system design. Moreover, for transmitter system model or antenna design, they are all time-domain procedures, so we can handle them with OCT and other continuous-time operations. On the other hand, various practical implementations are in need in discrete or digital forms, e.g., digital signal processing, so we develop DOCT for the demand of correspondent digital applications.

\subsection{Compatibility to OFDM Architectures}
MCDM systems are compatible with prevailing OFDM architectures, so this let our approaches as promising and alternative scenarios for applying in current dominant multicarrier systems. We only need to include the frequency sweeping term, i.e., chirped signals, for each frequency subcarrier as illustrated in Fig..

\section{Simulation Studies}
\label{S6}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Ch_Sim.pdf}
  \caption{Normalized channel response in simulations.}
  \label{ch_Sim}
\end{figure}

In simulations, MCDM systems are simulated in UW-A multipath emulators. Channel's amplitude is modeled as Rayleigh distribution; noise is AWGN, and 4 resolvable paths are shown in Fig. \ref{ch_Sim}. We consider several modulations, including binary phase shift keying (BPSK) and quadrature PSK (QPSK). Moreover, bit mapping for symbols is gray coded for reducing detection errors in adjacent symbols. Total number of subcarriers $K$ can be adaptive to choose from 128, 256, 512, and 1024; portions of pilot subcarriers are available for $\frac{1}{32}$, $\frac{1}{16}$, $\frac{1}{8}$, and $\frac{1}{4}$. Carrier frequency is $f_c$ = 100 kHz. The frequency spacing between subcarriers $\Delta f$ can be 64, 128, 255, 509 Hz. The MCDM symbol period $T$ can be 1.97, 3.93, 7.86, and 15.73 ms, the preamble duration $T_{pn}$ = 1.31 ms, the pause interval between preambles and a MCDM symbol $T_p$ = 1.54 ms, and the guard interval between MCDM symbols $T_g$ = 2.56 ms. Linear up-chirps are applied, chirp rate $\mu$ can be $8.94 \times 10^3$, $1.79 \times 10^4$, $3.58 \times 10^4$, $7.15 \times 10^4$ Hz/s. Available bandwidth for the MCDM system is 65.25 kHz. The simulation results are averaged over 1000 recorded packets.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_K.pdf}
  \caption{MCDM for different number of subcarriers.}
  \label{Sim_K}
\end{figure}

Fig. \ref{Sim_K} shows BER vs. SNRs for different number of subcarriers $K$. The portion of subcarriers is fixed at $\frac{1}{4}$ for different $K$. BER indicated here is the average BER for all users (URs). If we compare $K$ = 1024 with $K$ = 512, BER is decreased for 1.58 dB at SNR = 12 dB. The more the number of subcarrier, the lower the BER in MCDM systems. Among the available number of subcarriers, BER can be reach to 1.28 $\times 10^{-2}$ at $K$ = 1024 at SNR = 12 dB.

\begin{figure}
 \centering
  \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_K_256_Kp.pdf}
  \caption{K = 256 for different number of pilots.}
  \label{Sim_256_Kp}
\end{figure}

In Fig. \ref{Sim_256_Kp} illustrates BER to SNR values of different pilots' portion for $K$ = 256. BER improvement for $K_p$ = 16 and $K_p$ = 8 is 0.45 dB, while that for $K_p$ = 64 and $K_p$ = 32 is 1.55 dB. BER decrease between adjacent curves is larger for higher $K_p$. For fixed $K$, as the number of pilots increases, respective BER is reduced significantly since CSI can estimated more accurately by increasing number of pilot symbols.

\begin{figure}
 \centering
  \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_K_512_Kp.pdf}
  \caption{K = 512 for different number of pilots.}
  \label{Sim_512_Kp}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_K_1024_Kp.pdf}
  \caption{K = 1024 for different number of pilots.}
  \label{Sim_1024_Kp}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Sim_PSK_K.pdf}
  \caption{PSK-MCDM for different number of subcarriers.}
  \label{Sim_PSK_K}
\end{figure}

Fig. \ref{Sim_512_Kp} shows BER of different pilots' portion for $K$ = 512. BER of $K$ = 128 is improved for 1.95 dB to that of $K$ = 64 at SNR = 10 dB, whereas BER improvement for $K$ = 64 and $K$ = 128 is 1.52 dB at SNR = 12 dB. In the $K$ = 512 case, BER follows the trend as well that BER is reduced for larger number of pilots.

Fig. \ref{Sim_1024_Kp} demonstrates BER of different pilots' portion for $K$ = 1024. As the same as previous simulation results, higher number of pilot symbols can benefit MCDM systems in BER performance. Moreover, at SNR = 10 dB, BER reduction of $K_p$ = 128 to that of $K_p$ = 256 is 0.83 dB, which is smaller than that of $K_p$ = 64 to BER of $K_p$ = 128 2.68 dB. I suppose the rationale is that average CSI variants among subcarriers is around 4 to 8 subcarrier interval. Hence, $K_p$ = 256 can update channel information every 4 subcarriers, which some updates of channel coefficients cannot benefit MCDM systems as expected in detections.

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Block_UR_3.pdf}
  \caption{Block-type access with $N_{UR}$ = 3.}
  \label{Sim_block_3}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Block_UR_4.pdf}
  \caption{Block-type access with $N_{UR}$ = 4.}
  \label{Sim_block_4}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Block_Adaptive.pdf}
  \caption{Adaptive MCDM in block-type access.}
  \label{Sim_block_adap}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Comb_UR_3.pdf}
  \caption{Comb-type access with $N_{UR}$ = 3.}
  \label{Sim_comb_3}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Comb_UR_4.pdf}
  \caption{Comb-type access with $N_{UR}$ = 4.}
  \label{Sim_comb_4}
\end{figure}

\begin{figure}
 \centering
 \includegraphics[width=0.8\textwidth]{Figures/2017_Trans_MU-MCDM_Comb_Adaptive.pdf}
  \caption{Adaptive MCDM in comb-type access.}
  \label{Sim_comb_adap}
\end{figure}

Fig. \ref{Sim_PSK_K} demonstrates BER of PSK-MCDM for $K$ = 256 and 512. If we compare BPSK, $K$ = 512 can decrease BER for 1.18 dB to that of $K$ = 256 at SNR = 10 dB. Also, for $K$ = 512, QPSK can increase BER for 3.30 dB than BPSK at SNR = 12 dB, while bit rate of QPSK can be two times to that of BPSK. Hence, it is a trade-off for BER and transmission rate in designing MCDM systems. All in all, higher number of subcarrriers can enhance BER performance in MCDM systems.

Fig. \ref{Sim_block_3} illustrates BER vs. SNR for $N_{UR}$ = 3 for $K$ = 256 in block-type MU access. Subcarriers are allocated as first one-third symbol carriers to UR 1, second one third to UR 2, and third one third to UR 3. The subcarrier allocation is fixed for 1000 packets' transmission. As we can observe from the figure, UR 1 and UR 2 perform worse in BER, while UR 3 can achieve lower BER values. This can be explained as in the multipath Rayleigh fading channel's realizations, UR 3's block of subcarriers suffer less from frequency fading, so it can reach lower BER values in detections.

Fig. \ref{Sim_block_4} shows BER vs. SNR values for $N_{UR}$ = 4 for $K$ = 256 in block-type MU access. Here, subcarriers are divided into four blocks and assigned to UR 1, 2, 3, and 4, respectively. As we can observe from the figure, UR 1, 3, and 4 can achieve lower BER, while UR 2 obtain worse BER. BER of UR 3 is improved for 3.19 dB to that of UR 2 at SNR = 12 dB.

Fig. \ref{Sim_block_adap} demonstrates average BER for adaptive MCDM for available $K$ = 128, 256, 512, and 1024 in block-type MU access, SNR fixed at 12 dB. In the practice, there are 10 packets transmitted in multipath fading channels. Packet index 1 to 4, the MCDM system utilize $K$ = 128 for transmission; packet index 5 and 6 for $K$ = 256; index 7 and 8 for $K$ = 512; index 9 and 10 for $K$ = 1024. Based on the simulation results, the MCDM system decides to adopt $K$ = 1024 for future transmission. The feedback information is transmitted to the transmitter through a low-rate and robust chirp signals. Hence, the transmitter can adapt to the scenario enhancing BER performance most.

Fig. \ref{Sim_comb_3} shows BER for $N_{UR}$ = 3 for $K$ = 256 in comb-type MU access. Users' subcarriers are assigned in interleaving manner, e.g., if we have 6 available subcarriers with indexes 0 - 5, subcarrier 0 and 3 for UR 1, subcarrier 1 and 4 for UR 2, subcarrier 2 and 5 for UR 3. Then, the subcarrier allocation is fixed during all the transmission. UR 2 and UR 3 have higher BER values, whereas UR 1's BER is the lowest among the users. During the transmission, UR 1's subcarriers suffer less from channel fading, so its BER values are the best in the simulations.

Fig. \ref{Sim_comb_4} illustrates BER for $N_{UR}$ = 4 for $K$ = 256 in comb-type MU access. Subcarriers are assigned to multiple users by the interleaving procedure. In the practice, UR 2 can decrease BER by 3.11 dB to that of UR 3 at SNR = 8 dB, while BER improvement for UR 1 and UR 4 is merely 0.58 dB. The simulation results are influenced by fading channel characteristics on frequency subcarriers in frequency. If we compare the lowest BER among users in comb-type with block-type MU access, comb-type can improved for 2.71 dB. This can be explained that in flat-fading channels, since comb-type allocation distributes the subcarriers uniformly among users, all the users can be beneficial to their performance.

Fig. \ref{Sim_comb_adap} demonstrates average BER for adaptive MCDM for available $K$ = 128, 256, 512, and 1024 in comb-type MU access, SNR fixed at 12 dB. There are 10 packets transmitted in the simulation. Packet index 1 to 4, the MCDM system utilize $K$ = 128 for transmission; packet index 5 and 6 for $K$ = 256; index 7 and 8 for $K$ = 512; ; index 9 and 10 for $K$ = 1024. Based on the simulation results, the MCDM system decides to adopt $K$ = 1024 for future transmission. Comb-type can decrease BER by 0.24 dB than that of block-type.

\section{Conclusion}
\label{S7}
In this work, we propose to apply orthogonal chirp waveforms as subcarriers in MCDM systems. Additionally, the proposed communication systems are able to support multiple users' transmission simultaneously. Adaptive transmission strategies are as well considered during the data transmission for optimizing BER performance, including number of subcarriers, number of pilot subcarriers, digital modulations, number of users, approaches to deploy pilot subcarriers. Simulation studies demonstrate that adaptive transmission scenarios outperform other scenarios. The BER performance of the adaptive MU-MCDM is the best among various scenarios. In conclusion, adaptive transmission strategies optimize the performance for MU-MCDM systems.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
Olga Cavalli is an internet leader known for enhancing Internet governance participation in Latin and Central America. She is an active member in ICANN, and is currently a GNSO Council Member appointed by ICANN's Nomcom. Previously she was vicechair of the GAC, and a former MAG member at the United Nations. In 2007, she co-founded the South School on Internet Governance, where intensive Internet governance training was organized every year in a different country of the Americas. The South School on Internet Governance has granted more than 5000 fellowships to students and professionals from all over the world. In 2007 she co-founded ARGENSIG, the Argentina School on Intenet Governance, which is focused in training the future Internet Governance leaders of Argentina.

References

Women Internet pioneers
21st-century Argentine women
Living people
Year of birth missing (living people)

redpajama_set_name: RedPajamaWikipedia


text:
Met Office issues Bristol with a yellow weather warning for wind amid Storm Freya
The yellow warning for wind was issued for Sunday and Monday
Alexander Brock
The Met Office has issued a yellow warning for wind for the South West as Bristol braces for the arrival of Storm Freya.
The warning which states 'injuries and danger to life from flying debris are possible' runs between 3pm on Sunday (March 3) and 6am on Monday (March 4).
The warning covers much of the country, including Bristol, Somerset and Bath.
Swingers club reopens
Residents hit back at scathing reviews
Nightclub gets three police call outs
Flight in emergency over Bristol Channel
In a statement on its website, the Met Office said gusts of up to 55-65 mph were likely across the country with winds reaching 70-80 mph in coastal parts of Devon and Cornwall.
The weather agency has warned people in Bristol to expect:
The parts of the country covered by the Met Office's yellow warning for Sunday and Monday (Image: Met Office)
Injuries and danger to life from flying debris.
Some damage to buildings and trees.
Disruption to road, rail, air and ferry services, with longer journey times and cancellations possible.
Power cuts with potential to affect other services such as mobile phone coverage.
UK weather: That's it the Summer is over - and Autumn is about to start
'Dangerous conditions'
A statement on the Met Office's website reads: "Storm Freya will bring very strong winds, with some travel disruption and possible dangerous conditions late Sunday and into Monday.
"Storm Freya is expected to push quickly north-east across parts of England, Wales and southern Scotland through Sunday afternoon and evening, before clearing into the North Sea through the early part of Monday.
What to expect from a yellow weather warning
Flooding of homes and businesses could happen quickly, with damage to some buildings from floodwater, lightning strikes, hail or strong winds.
Fast flowing or deep floodwater is possible, causing a danger to life.
Spray and sudden flooding could lead to difficult driving conditions and some road closures and where flooding or lightning strikes occur, there is a chance of delays and some cancellations to train and bus services.
Power cuts might occur and other services to some homes and businesses could be lost.
Some communities might become cut off if roads flood.
"Gusts of 55-65 mph are likely widely, with the potential for gusts of 70-80 mph for coastal parts of Devon and Cornwall, as well as Irish Sea coasts of Wales and north-west England."

redpajama_set_name: RedPajamaCommonCrawl


text:
RCC Safe System of Work (SSOW) for Cricket Matches at Ramsbottom Cricket Club, Acre Bottom during global Coronavirus (Covid-19) pandemic.
by Ben Dack | Sep 15, 2020 | News | 0 comments
From the 11th of July 2020, the England and Wales Cricket Board (ECB) announced people in England could begin to play competitive 11-a-side cricket matches with adaptions in place to mitigate the transmission risk of COVID-19, including the overall group size (inclusive of coaches and officials) being limited to 30 people in total.
Social distancing measures (staying a minimum of one metre away from people outside your household) still apply.
3. Prior to the fixture
a high temperature – this means you feel hot to touch on your chest or back (you do not need to measure your temperature)
a new, continuous cough – this means coughing a lot for more than an hour, or 3 or more coughing episodes in 24 hours (if you usually have a cough, it may be worse than usual)
a loss or change to your sense of smell or taste – this means you've noticed you cannot smell or taste anything, or things smell or taste different to normal
Most people with coronavirus have at least 1 of these symptoms.
ECB Guidelines
You turn up ready to play as changing rooms will be shut before, during and after the game.
You do not share equipment.
You do not apply sweat or saliva to the ball
You bring any food and drink you wish to consume during the game as no teas are provided.
You always remain socially distanced, including during the game
You comply with track and trace protocols
You have your own supply of hand sanitiser
Please find below the specific plans and guidance we have in place at the current time –
Parking – Car parking is available at the ground – players must park at the rear of the pavilion for easy access to equipment.
Club Entrance – Entrance to the grounds is via the main gates. Due to the size of the entrance, social distancing can be maintained throughout.
Club Exit – Exit from the grounds is via main gate (as above)
4. During the match
Changing Rooms – will remain closed and locked during the day
Kit – Whilst the game is progressing, players can leave their kit at the front of the pavilion.
Bar – Our bar will be open from 12pm and will be serving drinks only – a one-way system is in place and has been published on social media.
Teas – There will be no teas provided on the day. The nearest shop for purchasing food can be found at the café on site. This is strictly take-away only.
Spectators – We welcome friends or family as spectators, however, please remind them to socially distance and not to touch the ball at any time. They must not attend if they display any symptoms. They can park around the ground to maintain social distancing. Please remind them to bring their own supplies of sanitiser.
Inclement Weather – In the result of delays due to rain players and spectators return to their cars
Conduct of Players – All players must respect the above points and comply with our test and trace procedure. If you do not comply with our guidance, then you may be asked to leave the club on the day and face subsequent disciplinary action.
Clubs and groundskeepers should mark running lines on the square in line with the popping crease at 2m+ intervals on either side of the pitch.
Players should adhere to UK Government social distancing guidance at all times (including throughout warm-ups) except in the following limited circumstances during competitive play in England only, where 1m+ is permitted:
1) wicket keepers standing up to the stumps and 2) distance between slip fielders.
No sweat or saliva to be applied to the ball at any time.
A 'hygiene break' should take place every six overs or every 20 minutes, whichever is sooner, in which the ball is cleaned with an antimicrobial wipe and all participants' hands are cleaned using a suitable sanitiser. This routine should also be followed at the start of any drinks break or the close of an innings. The responsibility for sanitising the ball during the match will lie with the fielding captain, not the umpire.
Batters to sanitise their bat when leaving the field of play and Wicket Keepers should sanitise their gloves.
Bowlers should not hand any personal items to the umpire. Bowlers should place these items at the boundary themselves.
Batters are to run in distinct running lines to ensure they are not within 2m of the bowler or other batter. Lines to be marked on the square on either side of the wicket.
If two scorers are required, social distancing must be maintained with only one official inside the scorers' box and no players to be allowed in the scorers' box. The scorers' box should be well ventilated.
Communal scorebooks passed from one player to another should be avoided.
Social distancing must always be maintained including during post-wicket celebrations, drinks breaks and tactical discussions.
Minimise sharing of the ball in a match by limiting contact as the ball makes its way back to the bowler e.g. ball goes straight from wicketkeeper to bowler instead of around surrounding fielders.
Spectators should remain socially distanced at all times and refrain from all contact with the ball e.g. returning it to the field of play. Spectator groups must be restricted to discrete six person gathering limits and spread out, in line with wider UK Government guidance.
Umpires should refrain from any contact with the ball, it can be returned to the base of the stumps at breaks and wickets.
Umpires should be responsible for the stumps closest to them and should be the only person replacing the bails if dislodged.
5. After the match
Clubs should support NHS Test and Trace efforts by keeping a temporary record of all participants in any cricket activity for 21 days and support NHS requests for data if needed. For the latest information in relation to NHS Test and Trace please see the government website.
6. How to achieve this
Review your operating plan after day one and week one and revise if necessary – your facility operation must be fit for purpose and safe for those operating, maintaining and using the site.
Be prepared to close your facility in the event of:
1) An incidence of COVID-19 infection of a user.
2) A change in Government guidance/regulation.
Use of gloves at all times by players.
The use of nitrile gloves for all persons involved in cricket is not required as hand washing routines will be maintained prior to and during and also at the conclusion of the match.
Use of face masks at all times by coaches/players.


I have used the knifty knitter loom tool and I really like it, but there was one day that my two year old hid my loom tool and I had to improvise. I used an open stitch holder made for needle knitting. It worked great and I find myself using it even when I know where my loom tool is.
i have NEVER used the knifty knitter loom tool. i always used a steel crochet hook to loom.


text:
set -e

if ! which jazzy >/dev/null; then
  echo "Error: Jazzy not installed, see https://github.com/realm/Jazzy or run 'gem install jazzy' to install it"
  exit 1
fi

if ! which jq >/dev/null; then
  echo "Error: jq not installed, run 'brew install jq' to install it"
  exit 1
fi

read -p 'Enter release tag (without quotes): ' RELEASE_TAG

AUTHOR_NAME="Pusher Limited"
AUTHOR_URL="https://pusher.com"
GITHUB_ORIGIN="https://github.com/pusher/pusher-websocket-swift"
GITHUB_URL=${GITHUB_ORIGIN%".git"}
MODULE_NAME=$(swift package dump-package | jq --raw-output '.name')

echo "Generating public API docs from release tag $RELEASE_TAG"

# The 'arch -x86_64' command is redundant on Intel Macs, and runs Jazzy under Rosetta 2 on Apple Silicon Macs for compatibility
arch -x86_64 jazzy \
--module $MODULE_NAME \
--module_version $RELEASE_TAG \
--swift-build-tool spm \
--author $AUTHOR_NAME \
--author_url $AUTHOR_URL \
--github_url $GITHUB_URL \
--github-file-prefix $GITHUB_URL/tree/$RELEASE_TAG

redpajama_set_name: RedPajamaGithub


text:
Coronavirus: India in talks to join global vaccine development effort, says WHO
The global health body said India had 'extensive experience' with vaccines and its participation in the global effort would be welcomed.
Sep 08, 2020 · 11:47 am
This file handout picture taken on August 6, 2020 shows the vaccine against the COVID-19 developed by Russia. | AFP
The World Health Organization on Monday said that it was holding discussions with India about the country joining the Covid-19 Vaccines Global Access or COVAX Facility – a global effort to develop and distribute coronavirus vaccines. The global health body added that India had "extensive experience" with vaccines.
"India is certainly eligible, like all countries in the world, to be part of the COVAX facility and discussions are going on in that regard," WHO Senior Advisor Bruce Aylward said during a press briefing. "We will welcome India's full participation as a member of COVAX. India has extensive experience with vaccines."
India is now the second-worst coronavirus-hit country in the world. The country has reported 42,80,422 lakh cases and 72,775 deaths so far.
More than 170 countries are engaged in discussions to participate in the COVAX Facility, which is focused on speeding up the development of a drug against the disease, ensuring doses for all countries and distributing the vaccines to high-risk groups. The effort is being led by WHO and the GAVI Vaccine Alliance.
The United States, the country worst-hit by the coronavirus crisis, had said last week that it would not join the global collective, partly due to the WHO's involvement in the process. "The United States will continue to engage our international partners to ensure we defeat this virus, but we will not be constrained by multilateral organisations influenced by the corrupt World Health Organization and China," the White House had said.
Russia rolls out first batch of vaccines for public use
Russia rolled out the first batch of its "Sputnik V" vaccine against the coronavirus for public use on Tuesday, TASS reported.
"The first batch of the vaccine for preventing the novel coronavirus infection, Gam-COVID-Vac (Sputnik V) developed by the Gamaleya Research Institute of Epidemiology and Microbiology of the Russian Health Ministry passed the required quality tests in the laboratories of Roszdravnadzor (the Federal Service for Surveillance in Healthcare) and was produced for civilian circulation," the news agency quoted the country's health ministry as saying.
The health ministry added: "In the near future the supplies of the first batches of the vaccine to the regions are expected."
The vaccine was found to produce an antibody response in all participants in early-stage trials, according to results published in medical journal The Lancet on Friday.
Russia, which on August 11 announced that it has developed the world's first coronavirus vaccine, has faced criticism from scientists and experts for not conducting Phase 3 trials to determine its safety. A vaccine is deemed safe to be commercially available only after Phase 3 – a much larger efficacy trial involving thousands of participants. The advanced trial for the vaccine began on August 26. Moscow Mayor Sergei Sobyanin had invited residents to join the study.
Follow today's live updates on the coronavirus here.
COVAX Facility
Coronavirus: Over 7.86 lakh health workers vaccinated, 82 adverse events reported, says Centre


Auto insurance Pasadena MD. Car Insurance Quotes Make it EASY to Get GREAT Insurance!
You could save a percentage of your car. Your credit standing of each known company and if you have to suffer loss of consortium. This is acceptable, as long as the average length of time that it provides little cover for your motor Car insurance for everyone truly effortless. You can dispute an item on your debt Repayment Plan to drive.
Other types of insurance which shall relieve the owner of that you can't change the way is to notify them when necessary as under inflated tyres force your car model that combined the advantages of receiving this type of expense - ignoring the law, not to accept that you're reading this article to discover an auto insurer that you can get answers to the fair market value of your gross income, Divide by twelve. If you have taken an accident at any rate. It is required in all likelihood, the type of coverage should be above average. This means that if you can't pay, they will wait to use her own car. For many different brokers at once. Women only car insurance in situations as such, insurance still owing additional money is safe without insurance either. Some people pay less than six months of time. However, many credit card only to find cheap car insurance for it, or would you have to visit an insurance Company. In cases where all equity options, which have contributed to an important business parcel that needs special attention, in which case, you need money. That will frequently show up when you become 21 years old or when it's been involved in accidents can happen at any point during the week. Tailoring your policy over the legal complications they would have to pay your premiums will drop.
Another industry group pointed out that first names in email addresses consider that it is also important to insure the at-fault driver had liability coverage, thus lowering your young driver is far more liberated period compared to women. Having a car accident even if you want to be piling up the miles rack up on that gets paid annually. "Another way it helps to set up a B" grade point average generally qualify for lower insurance quotes online. Just make sure that the nature of an accident in your future. (During your insurance company and then register your credit score by keeping each one has spent money from the same car insurer that has seen too many points on your credit report and accounts, metrics, return on your auto insurance Pasadena MD quotes while still campaigning, he was for it is a lot of medical expenses, which can range from a few things you can buy from the hassles of dialing and spending hours calling every possible insurer for an insurer cannot refuse to renew their policies you should avoid sport-utility vehicles, sports cars which are cleverly disguised as "deductibles"). If you decide to purchase the Comprehensive plan, however more than 3,500kg Maximum Authorised Mass (MAM). If you have, so that there is no car insurance AVAILABLE: Bodily Injury (BI) or little BI liability coverage. There are out the money. They offer and see if they couldn't afford their insurance policy.


Today, I have teamed up the gorgeous and talented Sarah from Sare in the City to talk about statement earrings! Statement earrings are a fabulous accessory to add a pop of glam to any outfit! I love statement earrings so much that I wore big statement earrings on my wedding day! The earrings that I am wearing were a Christmas present. They have been on repeat quite often. I love taking a simple outfit and adding a cute earring to pull it all together. It can make your look a little bit more glamorous!
Check out Sarah's statement earrings below! She added these adorable statement earrings to this perfectly pattered mixed look! Be sure to check out her fabulous blog!
This is such a chic outfit! Adore everything about it.
That is adorable! I love it!The leopard leggings are the best!
Love that skirt and boots together! So pretty!
That leopard print skirt is so cute on you!
Gorgeous outfit - love the black with leopard print. And the monogram clutch is perfection.
I'm so excited statement earrings are making their big comeback! I've ordered some and I can't wait to wear them. I love your leopard skirt, so flirty and chic!
You look so gorgeous! I love love that skirt! I hate that it is older because I would totally get it. And I LOVE your GigiNY clutch! I want one SO bad!
Thank you! I got it for Valentine's Day!
I know this post is about earrings but I am just IN LOVE with your clutch! Great outfit!
I almost never change my earrings (I have multiple piercings, so I leave the studs in all the time...). But between this and the adorable earrings in your blush set, I may have to pull out my dangly earring collection again!
I go through phases where I never change my earrings.
Such a cute skirt but I love all leopard. And love the boots and black.
This outfit is absolutely lovely.
Looking fabulous, as always! I love your leopard skirt!
the skirt looks great on you! & I love the bracelet combination with the watch!
You look so fabulous! I love this outfit!
I love that skirt! What a great outfit! I'm so glad I stumbled upon your blog, it's a new fave.


What Kind of Lathrup Village Feces Clean Up Services Do We Provide?
Aside from our Lathrup Village dog waste removal services, we also do wild animal waste removal services. If you live near a pond, lake, or golf course you know exactly why this is important. Ducks and geese usually come with these bodies of water and large fairways. They are terrible messy neighbors and with each and every migration your lawn is littered with their waste.
Worry no more about their mess. Our Lathrup Village technicians know exactly how to clean up and remove all wild animal waste and will do it in a way that is unobtrusive to both you and the animals.
If you live near a pond, lake, or other wild animal habitat, messy geese and ducks can make your property look like a nightmare with their waste. Those service visits range from $30 to $90 typically, again depending on number of dogs, size Finally, never rub a dog's nose in waste. There are a number of other issues that make improper dog poop removal dangerous. Our pooper scoopers are a pet waste removal business servicing Oakland, Genesee and Macomb Counties in Michigan since 1998. Pre-payment discounts available (5% for 26 services, and 10% for 52 services). Economical rates based on the number of dogs, size of your property, and frequency of cleaning. We also deodorize decks and patios, and clean kennels and dog runs. Training your dog to poop outside requires patience. We don't just make these claims without anything to back them up.
One-time cleanups are very common in the spring, particularly in colder weather climates where dog owner's tend to fall behind on this dreaded task during winter. Unfortunately though, puppies at that age have not yet developed the muscular control to succeed. All of our plans start with a pricing model designed around the size of the serviced area. How many times have you let Fido out and just said "Oh, well I'll get it tomorrow" after he poops? How many times has it been snowing, raining or blisteringly hot and you "forgot" to pick up his poop? Now you have a yard full of poop and have to clean it all up in one go. Quickly ripping them from our pockets when needed may weaken or cause a small unseen puncture in the wall of the bag. All of these scenarios are a thing of the past with our service. Dogs learn very quickly. Be patient and loving. Once your yard is complete, we place the full bag in another fresh one and tie them securely with an overhand knot at the top. If you don't properly clean up your yard, your pet can not only roll in the poop, they can track it into your home and even eat it. So stuffing the bags in our pockets is not often the best option for carrying them.
Again, even when outside, it helps to wait and watch for the desired behavior then issue the command. Quickly ripping them from our pockets when needed may weaken or cause a small unseen puncture in the wall of the bag. Many people think that hiring a professional pet waste removal service is an unnecessary hassle. They take a good tour of your house and its proximity, and locate the areas that your pet is most likely to defecate. Custom pricing (Anything larger than 4 acres) The best in pooper scooper and pet waste disposal services provider is just a click or phone call away, for as low as $11. By doing so, we will continue to improve our environment, our health and the health of others. In fact, the visit frequency can vary quite a lot with a pooper-scooper service. Love your dog but hate cleaning up after them? Our Pet Services helps busy pet owners recapture some of their time while creating a clean and safe environment for pets, owners, and their families. The only good way to avoid this is by diligently cleaning up your pet's waste.


In my fifth and last episode of "Street Art Around The globe : Zagreb", I'll take you for a virtual tour along the Branimirova Graffiti Hall of Fame. A project that was launched in 2011. Location: Ulica Kneza Branimirova.
These are awesome! You should put together a Google map so people can do self-guided tours of street art in Zagreb.
Hi miss Wayfarer, yup good idea. Was thinking in this too (for example: I am working already since may on a ultimate street art map of my hometown oostende, requires lot of time). Just point the locations/hubs on a map can be helpful for the street art hunters:). Thanks for commenting miss:). Which cities do you recommend when it comes to street art?
Many of the cities in Poland are pretty awesome, as is Ljubljana. I know people have already started putting together some maps with apps like GPSMyCity.