text:
\section{Introduction}
\label{sec:introduction}

In digital communication systems, various forms of distortion are ubiquitous, acting as the main limiting factor for information transmission. Those distortions that come with the propagation of signal, such as shadowing and multipath fading, have received extensive research since the earliest era of digital communications \cite{biglieri98:it}. The current paper, alternatively, concerns with the other category of distortions that come mainly with the engineering of transceivers. This category of distortions encompasses a number of models of practical importance, including the clipping or saturation of transmitted waveforms due to power amplifier nonlinearity, the analog-to-digital conversion ({\it i.e.}, quantization) of received samples, and others. Such distortions are difficult to eliminate, and indeed people may deliberately introduce them, for practical reasons like hardware cost reduction and energy efficiency improvement.

We can usually approximate the aforementioned transceiver distortions as memoryless deterministic functions. Those functions, however, are generally nonlinear operations and thus break down the linearity in Gaussian channels. From a pure information-theoretic perspective, nonlinearity may not impose fundamental difficulty to our conceptual understanding, since the channel capacity is still the maximum of mutual information between the channel input and the distorted channel output. From an engineering perspective, however, the general mutual information maximization problem is usually less satisfactory in generating insights, especially when such maximization problems are analytically difficult, or even intractable, for general nonlinear channel models.

There are a number of existing works that seek to characterize the information-theoretic behavior of nonlinear transceiver distortion, largely scattered in the literature. In \cite{ochiai02:com}, the authors examined the channel capacity of clipped orthogonal frequency-division multiplexing (OFDM) systems, with the key approximation that the distortion due to clipping acts as an additional Gaussian noise. Such an approximation originates from a theorem due to Bussgang \cite{bussgang52:rle}, which implies that the output process of a Gaussian input process through a memoryless distortion device is the sum of a scaled input process and a distortion process which is uncorrelated with the input process. Regarding Nyquist-sampled real Gaussian channels with output quantization, an earlier study \cite{viterbi79:book} examined the achievable mutual information as the signal-to-noise ratio (SNR) decreases toward zero. Specifically, the numerical study therein revealed that for a binary symmetric output quantizer, the ratio between the capacity per channel use (c.u.) and the SNR approaches $1/\pi$, and that for a uniform octal ({\it i.e.}, $8$-level) output quantizer, this ratio is no less than $0.475$. In \cite{singh09:com}, the authors further established some general results for Nyquist-sampled real Gaussian channels, asserting that with a $K$-level output quantization, the capacity is achieved by choosing no more than $(K + 1)$ input levels, and that with a binary symmetric output quantization the capacity is indeed achieved by using a binary symmetric input distribution. For $K > 2$, however, it is necessary to use intensive numerical methods like the cutting-plane algorithm to compute the capacity. The authors of \cite{mezghani08:isit} addressed the capacity of multiple-input-multiple-output block-Rayleigh fading channels with binary symmetric output quantization. In \cite{koch10:arxiv}, the authors went beyond the Nyquist-sampled channel model, demonstrating that the low-SNR capacity of a real Gaussian channel with binary symmetric output quantization, when sampled at twice the Nyquist rate, is higher than that when sampled at the Nyquist rate. In \cite{koch11:arxiv}, the authors proved that by using a binary asymmetric output quantizer design, it is possible to achieve the low-SNR asymptotic capacity without output quantization.

Recognizing the challenge in working with channel capacity directly, we take an alternative route that seeks to characterize achievable information rates for certain specific encoding/decoding scheme. As the starting point of our study, in the current paper we consider a real Gaussian channel with general transceiver distortion, and focus on the Gaussian codebook ensemble and the nearest-neighbor decoding rule. We use the so-called generalized mutual information (GMI) \cite{ganti00:it, lapidoth02:it} to characterize the achievable information rate. As a performance measure for mismatched decoding, GMI has proved convenient and useful in several other scenarios such as multipath fading channels \cite{lapidoth02:it}. Herein, in our exercise with GMI, we aim at providing key engineering insights into the understanding and design of transceivers with nonlinearity. The nature of our approach is somewhat similar to that of \cite{salz95:com}, where the authors addressed the decoder design with a finite resolution constraint, using a performance metric akin to cutoff rate that also derives from a random-coding argument.

The motivation for using the performance measure of GMI and the Gaussian codebook ensemble coupled with the nearest-neighbor decoding is two-fold. On one hand, such an approach enables us to obtain an array of analytical results that are both convenient and insightful, and bears an ``operational'' meaning in that the resulting GMI is achievable, by the specific encoding/decoding scheme whose implementation does not heavily depend on the nonlinear distortion model. On the other hand, Gaussian codebook ensemble is a reasonable model for approximating the transmitted signals in many modern communication systems, in particular, those that employ higher-order modulation or multicarrier techniques like OFDM\footnote{In the current paper we confine ourselves to the single-carrier real Gaussian channel model, and will treat multicarrier transmission with nonlinear distortion in a separate work.}; and the nearest-neighbor decoding rule is also a frequently encountered solution in practice which is usually easier to implement than maximum-likelihood decoding, for channels with nonlinear characteristics. Nevertheless, we need to keep in mind that compared with capacity, the performance loss of GMI due to the inherently suboptimal encoding/decoding scheme used may not be negligible.

The central result in the current paper is a GMI formula, taking the form of $(1/2) \log (1 + \mathrm{SNR}_e)$, for real Gaussian channels\footnote{For complex Gaussian channels we also have an analogous result; see Supplementary Material \ref{appendix:basic-model-complex}.} with general transceiver distortion. Here $\mathrm{SNR}_e$ depends on the nominal SNR and the transceiver nonlinearity, and we may interpret it as the ``effective SNR'', due to its apparent similarity with the role of SNR in the capacity formula for Gaussian channels without distortion. The parameter $\mathrm{SNR}_e$ thus serves as a single-valued performance indicator, based on which we can, in a unified fashion, analyze the behavior of given transceivers, compare different distortion models, and optimize transceiver design.

Applying the aforementioned general GMI formula to specific distortion models, we obtain an array of results that are of engineering relevance. First, when the nonlinear distortion occurs at the transmitter only, we show that the Bussgang decomposition, which represents a received signal as the sum of a scaled input signal part and a distortion part which is uncorrelated with the input signal, is consistent with the GMI-maximizing nearest-neighbor decoding rule. This result validates the Gaussian clipping noise approximation for transmit-side clipping, as followed by the authors of \cite{ochiai02:com}.

Second, we evaluate the GMI for Nyquist-sampled channels with output quantization. For binary symmetric quantization, we find that the low-SNR asymptotic GMI coincides with the channel capacity. This observation is somewhat surprising, since the GMI is with respect to a suboptimal input distribution, namely the Gaussian codebook ensemble. On the other hand, there exists a gap between high-SNR asymptotic GMI and the channel capacity, revealing the penalty of suboptimal input distribution when the effect of noise is negligible. For symmetric quantizers with more than two quantization levels, we formulate a quantizer optimization problem that yields the maximum GMI, and present numerical results for uniform and optimized quantizers. As an example of our results, we show that for octal quantizers, the low-SNR asymptotic GMI is higher than the known lower bound of channel capacity in the literature \cite{viterbi79:book}.

Finally, we explore the benefit of super-Nyquist sampling. Considering a real Gaussian channel with a bandlimited pulse-shaping function and with general memoryless output distortion, we obtain a formula for its GMI, when the channel output is uniformly sampled at $L$ times the Nyquist rate. We then particularize to the case of binary symmetric output quantization. We demonstrate through numerical evaluation that super-Nyquist sampling leads to benefit in terms of increased GMI over all SNR, for different values of $L$. In the low-SNR regime, the asymptotic GMI we obtain for $L = 2$ with a carefully chosen pulse-shaping function almost coincides with the known lower bound of channel capacity in the literature \cite{koch10:arxiv}. In the high-SNR regime, we make an interesting observation that, when the sampling rate is sufficiently high, the GMI becomes greater than one bit/c.u.. At first glance, this result is surprising since the output quantization is binary; however, it is in fact reasonable, because for each channel input symbol, there are multiple binary output symbols due to super-Nyquist sampling, and the amount of information carried by the Gaussian codebook ensemble exceeds one bit per input symbol.

We organize the remaining part of the paper as follows. Section \ref{sec:basic-model-real} describes the general Nyquist-sampled channel model and establishes the general GMI formula. Section \ref{sec:input-distortion} treats the scenario where only transmit-side distortion exists, revisiting the well-known decomposition of Bussgang's theorem. Section \ref{sec:1-bit} treats the channel model with binary symmetric output quantization. Section \ref{sec:multi-bit} treats symmetric output quantizers with more than two quantization levels. Section \ref{sec:super-nyquist} explores the benefit of super-Nyquist sampling. Finally Section \ref{sec:conclusions} concludes the paper. Auxiliary technical derivations and other supporting results are archived in the Supplementary Material.

\section{General Framework for Real-Valued Nyquist-Sampled Channels}
\label{sec:basic-model-real}

With Nyquist sampling, it loses no generality to consider a discrete-time channel model, with a sequence $\{\rvz_k\}$ of independent and identically distributed (i.i.d.) real Gaussian noise, {\it i.e.}, $\rvz_\cdot \sim \mathcal{N}(0, \sigma^2)$. The channel input symbols constitute a sequence $\{\rvx_k\}$. Without distortion, the received signal is $\rvy_\cdot = \rvx_\cdot + \rvz_\cdot$. However, the distortion may affect both the channel input and the channel output. A memoryless distortion, in general form, is a deterministic mapping $f(\cdot)$, which transforms a pair of channel input symbol and noise sample $(x, z)$ into a real number $f(x, z)$. Hence the channel observation at the decoder is
\begin{eqnarray}
\label{eqn:channel-distorted}
\rvw_k = f(\rvx_k, \rvz_k), \quad \mbox{for}\; k = 1, 2, \ldots, n,
\end{eqnarray}
where $n$ denotes the codeword length; see the illustration in Figure \ref{fig:model}. We note that, such a form of distortion can describe the case where the channel output $\rvy_\cdot = \rvx_\cdot + \rvz_\cdot$ is distorted, {\it i.e.}, $w = f(x, z) = f_o(x + z)$, or the case where the channel input $\rvx_\cdot$ is distorted by the transmitter, {\it i.e.}, $w = f(x, z) = f_i(x) + z$, or the case where both input and output are distorted, {\it i.e.}, $w = f(x, z) = f_o(f_i(x) + z)$.
\begin{figure}[h]
\epsfxsize=2in
\epsfclipon
\centerline{\epsffile{./figs/distortion_model.eps}}
\caption{Illustration of the general channel model with distortion.}
\label{fig:model}
\end{figure}

For transmission, the source selects a message $\rvm$ from $\mathcal{M} = \{1, 2, \ldots, \left\lfloor e^{nR} \right\rfloor\}$ uniformly randomly, and maps the selected message to a transmitted codeword, which is a length-$n$ real sequence, $\{\rvx_k(\rvm)\}_{k = 1}^n$. We restrict the codebook to be an i.i.d. $\mathcal{N}(0, \mathcal{E}_s)$ ensemble. That is, each codeword is a sequence of $n$ i.i.d. $\mathcal{N}(0, \mathcal{E}_s)$ random variables, and all the codewords are mutually independent. Such a choice of codebook ensemble satisfies the average power constraint $\frac{1}{n} \sum_{k = 1}^n \mathbf{E} \rvx_k^2(\rvm) \leq \mathcal{E}_s$. We thus define the nominal SNR as $\mathrm{SNR} = \mathcal{E}_s/\sigma^2$.

As is well known, when transceiver distortion is absent ({\it i.e.}, $w = y$), as the codeword length $n$ grows without bound, the Gaussian codebook ensemble achieves the capacity of the channel, $\frac{1}{2}\log(1 + \mathrm{SNR})$. In the following, we proceed to investigate the GMI when the channel experiences the memoryless nonlinear distortion $f(\cdot)$.

To proceed, we restrict the decoder to follow a nearest-neighbor rule, which, upon observing $\{w_k\}_{k = 1}^n$, computes for all possible messages, the distance metric,
\begin{eqnarray}
\label{eqn:nndec}
D(m) = \frac{1}{n}\sum_{k = 1}^n \left[ w_k - a x_k(m)\right]^2, \quad m \in \mathcal{M},
\end{eqnarray}
and decides the received message as $\hat{m} = \mathrm{arg}\min_{m \in \mathcal{M}} D(m)$. In (\ref{eqn:nndec}), the parameter $a$ is selected appropriately for optimizing the decoding performance. We note that, the nearest-neighbor decoder (with $a = 1$) coincides with the optimal (maximum-likelihood) decoder in the absence of distortion, but is in general suboptimal (mismatched) for the distorted channel (\ref{eqn:channel-distorted}).

In the subsequent development in this section, we characterize an achievable rate which guarantees that the average probability of decoding error decreases to zero as $n \rightarrow \infty$, for Gaussian codebook ensemble and the nearest-neighbor decoding rule, following the argument used in \cite{lapidoth02:it}. When we consider the average probability of decoding error averaged over both the message set and the Gaussian codebook ensemble, due to the symmetry in the codebook, it suffices to condition upon the scenario where the message $m = 1$ is selected for transmission.

With $m = 1$, we have
\begin{eqnarray}
\lim_{n \rightarrow \infty} D(1) &=& \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k = 1}^n \left[ \rvw_k - a \rvx_k(1)\right]^2 = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k = 1}^n \left[ f(\rvx_k, \rvz_k) - a \rvx_k(1)\right]^2\nonumber\\
&=& \mathbf{E}\left\{\left[f(\rvx, \rvz) - a\rvx\right]^2\right\} \quad \mbox{a.s.}
\end{eqnarray}
where $\rvx \sim \mathcal{N}(0, \mathcal{E}_s)$ and $\rvz \sim \mathcal{N}(0, \sigma^2)$, from the law of large numbers.

The exponent of the probability of decoding error is the GMI, given by
\begin{eqnarray}
I_{\mathrm{GMI}} = \sup_{\theta < 0} \left(\theta \mathbf{E}\left\{\left[f(\rvx, \rvz) - a\rvx\right]^2\right\} - \Lambda(\theta)\right),
\end{eqnarray}
where
\begin{eqnarray}
\Lambda(\theta) &=& \lim_{n \rightarrow \infty} \frac{1}{n} \Lambda_n(n \theta),\\
\label{eqn:Lambda_n}
\Lambda_n(n\theta) &=& \log \mathbf{E}\left\{\left.e^{n\theta D(m)}\right| \rvw_k, k = 1, \ldots, n \right\}, \quad \forall m \neq 1.
\end{eqnarray}
From Chernoff's bound and the union upper bounding technique, we see that as long as the information rate is less than $I_\mathrm{GMI}$, the average probability of decoding error decreases to zero as $n \rightarrow \infty$. Therefore, the GMI serves as a reasonable lower bound for the achievable information rate for a given codebook ensemble and a given decoding rule.

After the mathematical manipulation given in Supplementary Material \ref{appendix:proof-gmi-real}, we establish the following result.
\begin{prop}
\label{prop:gmi-real}
With Gaussian codebook ensemble and nearest-neighbor decoding, the GMI of the distorted channel (\ref{eqn:channel-distorted}) is
\begin{eqnarray}
\label{eqn:gmi-real}
I_\mathrm{GMI} = \frac{1}{2}\log \left(1 + \frac{\Delta}{1 - \Delta}\right),
\end{eqnarray}
where the parameter $\Delta$ is
\begin{eqnarray}
\Delta = \frac{\left\{ \mathbf{E}[ f(\rvx, \rvz) \rvx ]\right\}^2}{\mathcal{E}_s \mathbf{E}[f(\rvx, \rvz)]^2}.
\end{eqnarray}
The corresponding optimal choice of the decoding scaling parameter $a$ is $a_\mathrm{opt} = \mathbf{E}\left[ f(\rvx, \rvz) \rvx \right]/\mathcal{E}_s$.
\end{prop}

We readily see that $\Delta$ is the squared correlated coefficient between the channel input $\rvx$ and the distorted channel output $f(\rvx, \rvz)$, which is upper bounded by one, from Cauchy-Schwartz inequality. A larger value of $\Delta$ corresponds to a higher effective SNR.

When contrasted with the capacity of the undistorted channel, $\frac{1}{2}\log(1 + \mathrm{SNR})$, we can define the effective SNR of the distorted channel as $\mathrm{SNR}_e = \frac{\Delta}{1 - \Delta}$.

As an immediate verification, consider the undistorted channel $\rvw_\cdot = \rvx_\cdot + \rvz_\cdot$, for which we have $\Delta = {\mathcal{E}_s}/(\mathcal{E}_s + \sigma^2)$. Consequently, the effective SNR is $\mathrm{SNR}_e = \mathcal{E}_s/\sigma^2$, leading to the capacity of the undistorted channel.

It is perhaps worth noting that, the derivation of the GMI in fact does not require $\rvz_\cdot$ be Gaussian. Indeed, as long as $\{\rvz_k\}$ is an ergodic process and is independent of $\{\rvx_k\}$, the general result of Proposition \ref{prop:gmi-real} holds. However, for simplicity, in the current paper we confine ourselves to i.i.d. Gaussian noise, and do not pursue this issue further.

\subsubsection*{Remark on Antipodal Codebook Ensemble}

The foregoing analysis of GMI applies to any input distribution. Here, consider antipodal inputs, {\it i.e.}, $\rvx_k(m)$ takes $\sqrt{\mathcal{E}_s}$ and $-\sqrt{\mathcal{E}_s}$ with probability $1/2$, respectively. All the codeword symbols are mutually independent. Again, we consider a nearest-neighbor decoding rule, with distance metric in form of (\ref{eqn:nndec}). Following the same line of analysis as that for the Gaussian codebook ensemble, we have
\begin{eqnarray}
\label{eqn:gmi-antipodal}
I_\mathrm{GMI} = \sup_{t \in \mathbb{R}} \left(t \mathbf{E}[\rvx f(\rvx, \rvz)] - \mathbf{E}\log\cosh(t\sqrt{\mathcal{E}_s} f(\rvx, \rvz)) \right),
\end{eqnarray}
and the optimal value of $t$ should satisfy
\begin{eqnarray}
\label{eqn:gmi-antipodal-t}
\mathbf{E}\left[\sqrt{\mathcal{E}_s} f(\rvx, \rvz) \cdot \tanh(t \sqrt{\mathcal{E}_s} f(\rvx, \rvz))\right] = \mathbf{E}[\rvx f(\rvx, \rvz)].
\end{eqnarray}
Supplementary Material \ref{appendix:antipodal}. The evaluation of the GMI is usually more difficult than that for the Gaussian codebook ensemble.

\section{Channels with Transmit-Side Distortion: Bussgang Revisited}
\label{sec:input-distortion}

In this section, we briefly consider the scenario where only the channel input is distorted, {\it i.e.}, $w = f_i(x) + z$. Since $\rvx$ and $\rvz$ are independent, the optimal choice of the decoding scaling parameter becomes
\begin{eqnarray}
\label{eqn:scaling-input-distortion}
a_\mathrm{opt} = \frac{\mathbf{E}[(f_i(\rvx) + \rvz)\rvx]}{\mathcal{E}_s} = \frac{\mathbf{E} [\rvx f_i(\rvx)]}{\mathcal{E}_s}.
\end{eqnarray}
The resulting value of $\Delta$ is
\begin{eqnarray}
\Delta = \frac{\left\{\mathbf{E} [\rvx f_i(\rvx)]\right\}^2}{\mathcal{E}_s \left(\mathbf{E} [f_i(\rvx)]^2 + \sigma^2 \right)},
\end{eqnarray}
and the effective SNR is
\begin{eqnarray}
\mathrm{SNR}_e = \frac{\Delta}{1 - \Delta} = \frac{\left\{\mathbf{E} [\rvx f_i(\rvx)]\right\}^2}{\mathcal{E}_s \left(\mathbf{E} [f_i(\rvx)]^2 + \sigma^2 \right) - \left\{\mathbf{E} [\rvx f_i(\rvx)]\right\}^2}.
\end{eqnarray}

Inspecting $a_\mathrm{opt}$ in (\ref{eqn:scaling-input-distortion}), we notice that it leads to the following decomposition of $f_i(\rvx)$:
\begin{eqnarray}
f_i(\rvx) = a_\mathrm{opt} \rvx + \rvv,
\end{eqnarray}
where the distortion $\rvv$ is uncorrelated with the input $\rvx$. Recalling the Bussgang decomposition \cite{ochiai02:com}, we conclude that, when there is only transmit-side distortion, the optimal decoding scaling parameter in the nearest-neighbor decoding rule coincides with that suggested by Bussgang's theorem. Note that this conclusion does not hold in general when receive-side distortion exists.

\section{Channels with Binary Symmetric Output Quantization}
\label{sec:1-bit}

In this section, we consider the scenario where the channel output $\rvy = \rvx + \rvz$ passes through a binary symmetric hard-limiter to retain its sign information only. This is also called one-bit/mono-bit quantization/analog-to-digital conversion, and we can write it as $w = f(x, z) = \mathrm{sgn}(x + z)$.

For this scenario, we have
\begin{eqnarray}
\label{eqn:Delta-1-bit}
\Delta = \frac{\left\{\mathbf{E}[\rvx\cdot \mathrm{sgn}(\rvx + \rvz)]\right\}^2}{\mathcal{E}_s},
\end{eqnarray}
where we use the fact that the average output power $\mathbf{E}[\mathrm{sgn}(\rvx + \rvz)]^2$ is unity. Now in order to facilitate the evaluation of the expectation in the numerator in (\ref{eqn:Delta-1-bit}), we introduce the ``partial mean'' of the random variable $\rvx \sim \mathcal{N}(0, \mathcal{E}_s)$
\begin{eqnarray}
F(z) = \int_z^\infty \frac{x}{\sqrt{2\pi \mathcal{E}_s}} e^{-\frac{x^2}{2\mathcal{E}_s}} dx = \sqrt{\frac{\mathcal{E}_s}{2\pi}} \exp\left(-\frac{z^2}{2\mathcal{E}_s}\right),
\end{eqnarray}
which is an even function of $z \in (-\infty, \infty)$. We denote by $p_\rvx(x)$ and $p_\rvz(z)$ the probability density functions of $\rvx \sim \mathcal{N}(0, \mathcal{E}_s)$ and $\rvz \sim \mathcal{N}(0, \sigma^2)$, respectively, and proceed as
\begin{eqnarray}
\label{eqn:x-sgny}
&&\mathbf{E}[\rvx \cdot \mathrm{sgn}(\rvx + \rvz)] = \iint_{x + z > 0} x p_\rvx(x)p_\rvz(z) dx dz - \iint_{x + z < 0} x p_\rvx(x)p_\rvz(z) dx dz\nonumber\\
&=& 2\iint_{x + z > 0} x p_\rvx(x)p_\rvz(z) dx dz = 2\int_{-\infty}^\infty p_\rvz(z) F(-z) dz = \mathcal{E}_s \sqrt{\frac{2}{\pi (\mathcal{E}_s + \sigma^2)}}.
\end{eqnarray}
This leads to
\begin{eqnarray}
\Delta = \frac{\mathcal{E}_s^2 \frac{2}{\pi (\mathcal{E}_s + \sigma^2)}}{\mathcal{E}_s} = \frac{2\mathcal{E}_s}{\pi (\mathcal{E}_s + \sigma^2)},
\end{eqnarray}
and
\begin{eqnarray}
\mathrm{SNR}_e = \frac{\Delta}{1 - \Delta} = \frac{2\mathcal{E}_s}{(\pi - 2) \mathcal{E}_s + \pi \sigma^2}.
\end{eqnarray}
So we get the following asymptotic behavior:
\begin{itemize}
\item High SNR: When $\mathrm{SNR} = \mathcal{E}_s/\sigma^2 \rightarrow \infty$,
\begin{eqnarray}
\mathrm{SNR}_e &=& \frac{2}{\pi - 2} - \frac{2\pi}{(\pi - 2)^2}\frac{1}{\mathrm{SNR}} + o(\frac{1}{\mathrm{SNR}}),\\
I_\mathrm{GMI} &=& \frac{1}{2}\log \frac{\pi}{\pi - 2} - \frac{1}{\pi - 2} \frac{1}{\mathrm{SNR}} + o(\frac{1}{\mathrm{SNR}}).
\end{eqnarray}
\item Low SNR: When $\mathrm{SNR} \rightarrow 0$,
\begin{eqnarray}
\mathrm{SNR}_e &=& \frac{2}{\pi} \mathrm{SNR} - \frac{2(\pi - 2)}{\pi^2} \mathrm{SNR}^2 + o(\mathrm{SNR}^2),\\
I_\mathrm{GMI} &=& \frac{1}{\pi} \mathrm{SNR} - \frac{\pi - 1}{\pi^2} \mathrm{SNR}^2 + o(\mathrm{SNR}^2).
\end{eqnarray}
\end{itemize}

We make two observations. First, at high SNR, the GMI converges to $0.7302$ bits/c.u., strictly less than the limit of the channel capacity, $1$ bit/c.u., revealing that the suboptimal Gaussian codebook ensemble leads to non-negligible penalty when the effect of distortion is dominant. Second, at low SNR, the ratio between the GMI and the SNR converges to $1/\pi$, and thus asymptotically coincides with the behavior of the channel capacity \cite{viterbi79:book}. Intuitively, this is because in the low-SNR regime the effect of noise is dominant, and thus the channel is approximately still Gaussian. In Figure \ref{fig:binary-adc-nyquist} we plot the GMI $I_\mathrm{GMI}$ and the channel capacity $C = 1 - H_2(Q(\sqrt{\mathcal{E}_s/\sigma^2}))$ \cite{singh09:com} versus $\mathrm{SNR}$. The different behaviors of the GMI in the two regimes are evident in the figure.
\begin{figure}[ht]
\centering
\includegraphics[width=3.5in]{./figs/binary_adc_nyquist.eps}
\caption{The GMI and the channel capacity of the real Gaussian channel with binary symmetric output quantization.}
\label{fig:binary-adc-nyquist}
\end{figure}

\section{Channels with Multi-Bit Output Quantization}
\label{sec:multi-bit}

In this section, we continue the exploration of output quantization and consider specifically the scenario where the channel output $\rvy$ passes through a $2M$-level symmetric quantizer, as
\begin{eqnarray}
w = f(x + z) = r_i \cdot \mathrm{sgn}(x + z) \quad \mbox{if}\; |x + z| \in [\alpha_{i - 1}, \alpha_i),
\end{eqnarray}
for $i = 1, \ldots, M$, where $\alpha_0 = 0 < \alpha_1 < \ldots < \alpha_M = \infty$. The parameters include the reconstruction points $\{r_1, \ldots, r_M\}$, and the quantization thresholds $\{\alpha_1, \ldots, \alpha_{M - 1}\}$. Note that with $2M$ levels, the quantizer bit-width is $(\log_2 M + 1)$ bits.

For a $2M$-level symmetric quantizer, we can evaluate that (see Supplementary Material \ref{appendix:2M-ADC-Delta-proof})
\begin{eqnarray}
\label{eqn:2M-ADC-Delta-1}
\mathbf{E}[f(\rvx + \rvz)]^2 = 2 \sum_{i = 1}^M r_i^2 \left[Q\left(\frac{\alpha_{i - 1}}{\sqrt{\mathcal{E}_s + \sigma^2}}\right) - Q\left(\frac{\alpha_i}{\sqrt{\mathcal{E}_s + \sigma^2}}\right)\right],
\end{eqnarray}
where the Q-function is $Q(z) = \frac{1}{\sqrt{2\pi}} \int_z^\infty e^{-x^2/2} dx$, and
\begin{eqnarray}
\label{eqn:2M-ADC-Delta-2}
\mathbf{E}[f(\rvx + \rvz) \rvx] = \mathcal{E}_s\sqrt{\frac{2}{\pi(\mathcal{E}_s + \sigma^2)}}\sum_{i = 1}^M r_i \left[e^{-\frac{\alpha_{i - 1}^2}{2(\mathcal{E}_s + \sigma^2)}} - e^{-\frac{\alpha_i^2}{2(\mathcal{E}_s + \sigma^2)}}\right].
\end{eqnarray}
To further simplify the notation, define $\tilde{Q}(z) = \frac{1}{2\sqrt{\pi}} \int_0^z (-\log x)^{-1/2} dx$ for $z \in [0, 1)$,\footnote{We have $\tilde{Q}(z) = Q(\sqrt{-2 \log z}) = (1/2)\cdot \mathrm{erfc}(\sqrt{-\log z})$.} and introduce $t_i = e^{-\frac{\alpha_i^2}{2(\mathcal{E}_s + \sigma^2)}}$ for $i = 0, 1, \ldots, M$ with $t_0 = 1 > t_1 > \ldots > t_M = 0$. We thus can rewrite
\begin{eqnarray}
\mathbf{E}[f(\rvx + \rvz)]^2 &=& 2\sum_{i = 1}^M r_i^2 [\tilde{Q}(t_{i - 1}) - \tilde{Q}(t_i)],\\
\mathbf{E}[f(\rvx + \rvz) \rvx] &=& \mathcal{E}_s\sqrt{\frac{2}{\pi(\mathcal{E}_s + \sigma^2)}} \sum_{i = 1}^M r_i (t_{i - 1} - t_i).
\end{eqnarray}
These lead to
\begin{eqnarray}
\label{eqn:delta-multi-bit}
\Delta = \frac{\mathcal{E}_s}{\pi (\mathcal{E}_s + \sigma^2)} \frac{\left[\sum_{i = 1}^M r_i (t_{i - 1} - t_i) \right]^2}{\sum_{i = 1}^M r_i^2 [\tilde{Q}(t_{i - 1}) - \tilde{Q}(t_i)]}.
\end{eqnarray}
In (\ref{eqn:delta-multi-bit}), the second term is independent of the SNR, and can be optimized separately. Let us denote this term by $K_{\underline{r}, \underline{t}}$, and write $\Delta = \frac{\mathcal{E}_s K_{\underline{r}, \underline{t}}}{\pi (\mathcal{E}_s + \sigma^2)}$. We consequently have the following effective SNR:
\begin{eqnarray}
\label{eqn:eff-snr-mbit}
\mathrm{SNR}_e = \frac{K_{\underline{r}, \underline{t}} \mathcal{E}_s}{(\pi - K_{\underline{r}, \underline{t}}) \mathcal{E}_s + \pi \sigma^2}.
\end{eqnarray}
\begin{itemize}
\item High SNR: When $\mathrm{SNR} \rightarrow \infty$,
\begin{eqnarray}
\mathrm{SNR}_e &=& \frac{K_{\underline{r}, \underline{t}}}{\pi - K_{\underline{r}, \underline{t}}} - \frac{K_{\underline{r}, \underline{t}}\pi}{(\pi - K_{\underline{r}, \underline{t}})^2}\frac{1}{\mathrm{SNR}} + o(\frac{1}{\mathrm{SNR}}),\\
I_\mathrm{GMI} &=& \frac{1}{2}\log \frac{\pi}{\pi - K_{\underline{r}, \underline{t}}} - \frac{K_{\underline{r}, \underline{t}}}{2(\pi - K_{\underline{r}, \underline{t}})} \frac{1}{\mathrm{SNR}} + o(\frac{1}{\mathrm{SNR}}).
\end{eqnarray}
\item Low SNR: When $\mathrm{SNR} \rightarrow 0$,
\begin{eqnarray}
\mathrm{SNR}_e &=& \frac{K_{\underline{r}, \underline{t}}}{\pi} \mathrm{SNR} - \frac{K_{\underline{r}, \underline{t}}(\pi - K_{\underline{r}, \underline{t}})}{\pi^2} \mathrm{SNR}^2 + o(\mathrm{SNR}^2),\\
I_\mathrm{GMI} &=& \frac{K_{\underline{r}, \underline{t}}}{2\pi} \mathrm{SNR} - \frac{K_{\underline{r}, \underline{t}}(\pi - K_{\underline{r}, \underline{t}}/2)}{2\pi^2} \mathrm{SNR}^2 + o(\mathrm{SNR}^2).
\end{eqnarray}
\end{itemize}

It is thus apparent that the value of $K_{\underline{r}, \underline{t}}$ determines the system performance, for all SNR. We hence seek to maximize
\begin{eqnarray}
K_{\underline{r}, \underline{t}} = \frac{\left[\sum_{i = 1}^M r_i (t_{i - 1} - t_i) \right]^2}{\sum_{i = 1}^M r_i^2 [\tilde{Q}(t_{i - 1}) - \tilde{Q}(t_i)]},
\end{eqnarray}
where $t_0 = 1 > t_1 > \ldots > t_M = 0$ and $r_i \geq 0$ for all $i = 1, \ldots, M$.

Taking the partial derivatives of $K_{\underline{r}, \underline{t}}$ with respect to $r_i$, $i = 1, \ldots, M$, and enforcing them to vanish, we have that the following set of equations needs to hold for maximizing $K_{\underline{r}, \underline{t}}$,
\begin{eqnarray}
r_i = \frac{t_{i - 1} - t_i}{\tilde{Q}(t_{i - 1}) - \tilde{Q}(t_i)} \frac{\sum_{j = 1}^M r_j^2 [\tilde{Q}(t_{j - 1}) - \tilde{Q}(t_j)]}{\sum_{j = 1}^M r_j (t_{j - 1} - t_j)}, \quad i = 1, \ldots, M.
\end{eqnarray}
Substituting these $\{r_i\}$ into $K_{\underline{r}, \underline{t}}$ and simplifying the resulting expression, we obtain
\begin{eqnarray}
K_{\underline{t}} = \max_{\underline{r}} K_{\underline{r}, \underline{t}} = \sum_{i = 1}^M \frac{(t_{i - 1} - t_i)^2}{\tilde{Q}(t_{i - 1}) - \tilde{Q}(t_i)}.
\end{eqnarray}
That is, the optimal quantizer design should solve the following maximization problem:
\begin{eqnarray}
\label{eqn:maximization-Kt}
\max_{\underline{t}} \sum_{i = 1}^M \frac{(t_{i - 1} - t_i)^2}{\tilde{Q}(t_{i - 1}) - \tilde{Q}(t_i)},\quad \mbox{s.t.}\quad t_0 = 1 > t_1 > \ldots > t_M = 0.
\end{eqnarray}

{\it Example:} Fine quantization, $\max_{i = 1, \ldots, M} (t_{i - 1} - t_i) \rightarrow 0$

In this case, the following approximation becomes accurate:
\begin{eqnarray}
\frac{\tilde{Q}(t_{i - 1}) - \tilde{Q}(t_i)}{t_{i - 1} - t_i} \approx \tilde{Q}^\prime (t_{i - 1}), \quad \forall i = 1, \ldots, M.
\end{eqnarray}
So the resulting $K_{\underline{t}}$ behaves like
\begin{eqnarray}
K_{\underline{t}} &=& \sum_{i = 1}^M \frac{(t_{i - 1} - t_i)^2}{\tilde{Q}(t_{i - 1}) - \tilde{Q}(t_i)} \rightarrow \int_0^1 \frac{1}{\tilde{Q}^\prime(t)} dt\nonumber\\
&=& 2\sqrt{\pi} \int_0^1 \sqrt{-\log t} dt = 2\sqrt{\pi} \int_{-\infty}^\infty y^2 e^{-y^2}dy = \pi.
\end{eqnarray}
Therefore, as the quantization goes fine asymptotically, the effective SNR as given by (\ref{eqn:eff-snr-mbit}) approaches the actual SNR, and thus the performance loss due to quantization eventually diminishes.

{\it Example:} 4-level quantization, $M = 2$

In this case, there is only one variable, $t = t_1$, to optimize. The maximization problem becomes
\begin{eqnarray}
\max_{t \in (0, 1)} \frac{(1 - t)^2}{1/2 - \tilde{Q}(t)} + \frac{t^2}{\tilde{Q}(t)}.
\end{eqnarray}
A numerical computation immediately gives $\max_{t \in (0, 1)} K_t = 2.7775$, and interestingly, the maximizing $t = 0.618$ is the golden ratio.

{\it Example:} Uniform quantization

In practical systems, uniform quantization is common, in which the thresholds satisfy $\alpha_i = i \sqrt{2(\mathcal{E}_s + \sigma^2)\alpha}$ for $i = 0, 1, \ldots, M - 1$, and $\alpha_M = \infty$, where $\alpha > 0$ is a parameter for optimization. These thresholds lead to
\begin{eqnarray}
K_{\underline{t}} = \sum_{i = 1}^{M - 1} \frac{\left[e^{-(i - 1)^2 \alpha} - e^{-i^2 \alpha}\right]^2}{Q(\sqrt{2\alpha} (i - 1)) - Q(\sqrt{2\alpha} i)} + \frac{e^{-2(M - 1)^2 \alpha}}{Q(\sqrt{2\alpha}(M - 1))},
\end{eqnarray}
which can be further maximized over $\alpha > 0$.

In Table \ref{tab:uniform}, we list the numerical results for optimizing $K_{\underline{t}}$ over $\alpha$, up until $M = 8$.
\begin{table}[ht]
\vspace{0.1in}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
  \hline

  $M$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
  \hline
  $\max_\alpha K_{\underline{t}}$ & 2.7725 & 2.9569 & 3.0291 & 3.0651 & 3.0858 & 3.0989 & 3.1077 \\
  \hline
  optimal $\alpha$ & 0.481 & 0.253 & 0.159 & 0.111 & 0.082 & 0.064 & 0.051 \\
  \hline
\end{tabular}
\vspace{0.1in}
\caption{Table of performance for optimized uniform $2M$-level symmetric output quantization.}
\label{tab:uniform}
\end{table}

{\it Example:} $t$-uniform quantization

An alternative quantizer design is to let the values of $\underline{t}$ be uniformly placed within $[0, 1]$, {\it i.e.}, $t_i = (M - i)/M$, for $i = 0, 1, \ldots, M$. This quantization leads to
\begin{eqnarray}
K_{\underline{t}} = \frac{1}{M^2} \sum_{i = 1}^M \frac{1}{\tilde{Q}(1 - (i - 1)/M) - \tilde{Q}(1 - i/M)}.
\end{eqnarray}

In Table \ref{tab:t-uniform}, we list the numerical results of $K_{\underline{t}}$ for $t$-uniform quantizers, up until $M = 8$. We notice that the $t$-uniform quantization is consistently inferior to the optimized uniform quantization.
\begin{table}[ht]
\vspace{0.1in}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
  \hline

  $M$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
  \hline
  $K_{\underline{t}}$ & 2.7488 & 2.9267 & 3.0011 & 3.0404 & 3.0642 & 3.0798 & 3.0908 \\
  \hline
\end{tabular}
\vspace{0.1in}
\caption{Table of performance for $t$-uniform $2M$-level symmetric output quantization.}
\label{tab:t-uniform}
\end{table}

{\it Example:} Optimal quantization

We can also develop program to numerically solve the optimization problem (\ref{eqn:maximization-Kt}). In Table \ref{tab:optimal}, we list the results, up until $M = 8$. We also list the value of the optimal $t_1$, from which we can recursively compute the whole optimal $\underline{t}$ vector, through enforcing the partial derivatives $\partial{K_{\underline{t}}}/{\partial t_i}$ to vanish for $i = 2, \ldots, M-1$ progressively.
\begin{table}[ht]
\vspace{0.1in}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
  \hline

  $M$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
  \hline
  $\max_{\underline{t}} K_{\underline{t}}$ & 2.7725 & 2.9595 & 3.0330 & 3.0695 & 3.0902 & 3.1032 & 3.1117 \\
  \hline
  optimal $t_1$ & 0.618 & 0.805 & 0.880 & 0.922 & 0.943 & 0.958 & 0.967 \\
  \hline
\end{tabular}
\vspace{0.1in}
\caption{Table of performance for optimal $2M$-level symmetric output quantization.}
\label{tab:optimal}
\end{table}

From the numerical results in the above examples, we observe that the GMI may be fairly close to the channel capacity at low SNR. For example, with the optimal octal quantizer ($M = 4$), the low-SNR GMI scales with SNR like $0.4827 \cdot \mathrm{SNR}$ bits/c.u., which is better than the known lower bound $0.475 \cdot \mathrm{SNR}$ bits/c.u. in the literature \cite{viterbi79:book}. In Figure \ref{fig:opt-adc-nyquist} we plot the GMI $I_\mathrm{GMI}$ achieved by the optimal quantizers, for $M = 2, 3, \ldots, 8$. For comparison we also plot in dash-dot curve the capacity $(1/2)\log_2(1 + \mathrm{SNR})$ of undistorted channels. We can roughly assess that, with $M = 4$ ({\it i.e.}, 3-bit quantization), the performance gap between the GMI and the undistorted channel capacity is mild up until $\mathrm{SNR} \approx 10$ dB; and with $M = 8$ ({\it i.e.}, 4-bit quantization), the performance gap is mild up until $\mathrm{SNR} \approx 15$ dB. Compared with the numerically evaluated capacity for 2/3-bit quantization in \cite{singh09:com}, we see that using the Gaussian codebook ensemble and the nearest-neighbor decoding rule induce a 15-25\% rate loss at high SNR. Comparing Tables \ref{tab:uniform} and \ref{tab:optimal}, we further notice that the performance loss due to using uniform quantization is essentially negligible.
\begin{figure}[ht]
\centering
\includegraphics[width=3.5in]{./figs/opt_adc_nyquist.eps}
\caption{The GMI achieved by optimal $2M$-level quantizers, for $M = 2, 3, \ldots, 8$.}
\label{fig:opt-adc-nyquist}
\end{figure}

\subsubsection*{Remark on Possible Connection with Capacity per Unit Cost}

For a given $2M$-level symmetric quantizer, we can evaluate the channel capacity per unit cost (symbol energy in our context) by optimizing a single nonzero input symbol, $x$ (see \cite{verdu90:it}). Without loss of generality, we let $x > 0$ and the noise variance $\sigma^2$ be unity. Then the capacity per unit cost can be evaluated as
\begin{eqnarray}
\label{eqn:cpuc}
\sup_{x > 0} \frac{1}{x^2} \sum_{i = 1}^M \left[(Q(\alpha_{i - 1} - x) - Q(\alpha_i - x))\log\frac{Q(\alpha_{i - 1} - x) - Q(\alpha_i - x)}{Q(\alpha_{i - 1}) - Q(\alpha_i)} \right.\nonumber\\
\left.+ (Q(\alpha_{i - 1} + x) - Q(\alpha_i + x))\log\frac{Q(\alpha_{i - 1} + x) - Q(\alpha_i + x)}{Q(\alpha_{i - 1}) - Q(\alpha_i)}\right].
\end{eqnarray}
With some manipulations, we find that $K_{\underline{t}}/(2\pi)$ is exactly the limit value of the term in (\ref{eqn:cpuc}) as $x \rightarrow 0$.\footnote{This is also half of the Fisher information for estimating $\rvx = 0$ from the quantized channel output $\rvw$ \cite{verdu90:it}.} Therefore, only if the capacity per unit cost (\ref{eqn:cpuc}) is achieved by $x \rightarrow 0$, the GMI coincides with the channel capacity in the low-SNR limit. Unfortunately, as revealed by our numerical experiments, this condition does not generally hold for all possible symmetric quantizers.

\section{Super-Nyquist Output Sampling}
\label{sec:super-nyquist}

In this section, we examine the scenario where we sample the channel output at a rate higher than the Nyquist rate, and investigate the benefit of increased  sampling rates in terms of the GMI.

We start with a continuous-time baseband model in which the transmitted signal is
\begin{eqnarray}
x(t) = \frac{1}{\sqrt{2W}} \sum_{k = 1}^n \rvx_k g\left(t - \frac{k}{2W}\right),
\end{eqnarray}
where $g(\cdot)$ is a pulse function with unit energy and is band limited within $W$ Hz. In analysis, a commonly used pulse function is the sinc function $g(t) = \sqrt{2W} \mathrm{sinc}(2Wt)$ with $\mathrm{sinc}(t) = \sin(\pi t)/(\pi t)$, which vanishes at the Nyquist sampling time instants $t = \{k/(2W)\}_{k = -\infty}^\infty$. The channel input is a sequence of independent $\mathcal{N}(0, \mathcal{E}_s)$ random variables $\{\rvx_k\}_{k = 1}^n$. With additive white Gaussian noise $z(t)$, the received signal is
\begin{eqnarray}
y(t) = x(t) + z(t).
\end{eqnarray}
We assume that $z(t)$ is band-limited within $W$ Hz, with in-band two-sided power spectral density $\sigma^2/2$. So the autocorrelation function of $z(t)$ is $K_z(\tau) = \frac{\sigma^2}{2} \mathrm{sinc}(2Wt)$.

We consider a uniform sampler, which samples the channel output $y(t)$ at $L$ times the Nyquist rate. For the $k$-th input symbol, the sampling time instants thus are
\begin{eqnarray}
\label{eqn:sampling-time}
t = \left\{\frac{k}{2W} + \frac{l}{2WL} - \tau_L\right\}_{l = 0}^{2(L - 1)}.
\end{eqnarray}
Here, $\tau_L$ is a constant offset to ensure that the sampling times are symmetric with respect to the center of the $k$-th input symbol pulse; for example, $\tau_1 = 0$ (Nyquist sampling), $\tau_2 = 1/(4W)$, $\tau_3 = 1/(3W)$, $\tau_4 = 3/(8W)$... Generally, $\tau_L = \frac{L - 1}{L} \frac{1}{2W}$. Thus we can rewrite (\ref{eqn:sampling-time}) as
\begin{eqnarray}
t = \frac{1}{2W}\left\{k + \frac{l}{L}\right\}_{l = -L+1}^{L-1}.
\end{eqnarray}
Denote the output samples by $\{\rvy_{k, l}\}$ with $\rvy_{k, l} = y(t_{k, l})$ where $t_{k, l} = \frac{1}{2W}(k + l/L)$. The samples pass through a nonlinear distortion device, so that the observed samples are $\rvw_{k, l} = f(\rvy_{k, l})$.

Let us generalize the nearest-neighbor decoding rule in Section \ref{sec:basic-model-real} as follows. For all possible messages, the decoder computes the distance metrics,
\begin{eqnarray}
D(m) = \frac{1}{n}\sum_{k = 1}^n \sum_{l = -L+1}^{L-1} \xi_l [w_{k, l} - a_l x_k(m)]^2, \quad m \in \mathcal{M},
\end{eqnarray}
where $\{\xi_l\}_{l = -L+1}^{L-1}$ and $\{a_l\}_{l = -L+1}^{L-1}$ are weighting coefficients, and decides the received message as $\hat{m} = \mathrm{arg}\min_{m \in \mathcal{M}} D(m)$. We then note that
\begin{eqnarray*}
&&\sum_{l = -L+1}^{L-1} \xi_l [w_{k, l} - a_l x_k(m)]^2 = \sum_{l = -L+1}^{L-1} \xi_l w_{k, l}^2 - 2 x_k(m) \sum_{l = -L+1}^{L-1} \xi_l a_l w_{k, l} + x_k^2(m) \sum_{l = -L+1}^{L-1} \xi_l a_l^2\nonumber\\
&=& \left(\sum_{l = -L+1}^{L-1} \xi_l a_l^2\right)\cdot \left[x_k(m) - \frac{\sum_{l = -L+1}^{L-1} \xi_l a_l w_{k, l}}{\sum_{l = -L+1}^{L-1} \xi_l a_l^2}\right]^2 + \left[\sum_{l = -L+1}^{L-1} \xi_l w_{k, l}^2 - \frac{\left(\sum_{l = -L+1}^{L-1} \xi_l a_l w_{k, l}\right)^2}{\sum_{l = -L+1}^{L-1} \xi_l a_l^2}\right].
\end{eqnarray*}
Therefore, without loss of generality, we may consider the simplified nearest-neighbor decoding distance metric
\begin{eqnarray}
D(m) = \frac{1}{n} \sum_{k = 1}^n \left[\sum_{l = -L+1}^{L-1} \beta_l w_{k, l} - x_k(m)\right]^2,
\end{eqnarray}
for which the tunable weighting coefficients are $\underline{\beta} = \{\beta_l\}_{l = -L+1}^{L-1}$.

Following the same procedure as that in Section \ref{sec:basic-model-real} for the Nyquist-sampled channel model, we first examine the limit of $D(1)$ assuming that the message $m = 1$ is sent. Since the channel input symbols $\rvx_\cdot$ are i.i.d. and the noise process is wide-sense stationary, the observed samples $\rvw_{k, l}$ constitute an ergodic process.\footnote{We note that the transmission of a codeword, $\{\rvx_k\}_{k = 1}^n$, is finite-length. In order to meet the ergodicity condition, we may slightly modify the model by appending $\{\rvx_k\}_{k = -\infty}^0$ and $\{\rvx_k\}_{k = n + 1}^\infty$, which consist of i.i.d. $\mathcal{N}(0, \mathcal{E}_s)$ random variables as additional interference, to the transmitted codeword.} Consequently, we have
\begin{eqnarray}
\lim_{n \rightarrow \infty} D(1) = \mathbf{E} \left[\sum_{l = -L+1}^{L-1} \beta_l \rvw_{0, l} - \rvx_0\right]^2 \quad \mbox{a.s.}
\end{eqnarray}
On the other hand, for any $m \neq 1$, we have
\begin{eqnarray}
\frac{1}{n} \Lambda_n(n\theta)
&=& \frac{1}{n} \log \mathbf{E} \left\{\left.e^{\theta \sum_{k = 1}^n \left[\sum_{l = -L+1}^{L - 1} \beta_l \rvw_{k, l} - \rvx_k(m)\right]^2}\right| \rvw_{k, l}, k = 1, \ldots, n, l = -L+1,
\ldots, L - 1\right\}\nonumber\\
&=& \frac{\theta}{1 - 2\theta \mathcal{E}_s} \frac{1}{n} \sum_{k = 1}^n \left[\sum_{l = -L+1}^{L - 1} \beta_l \rvw_{k, l}\right]^2 - \frac{1}{2} \log(1 - 2\theta \mathcal{E}_s)\nonumber\\
&\rightarrow& \frac{\theta}{1 - 2\theta \mathcal{E}_s} \mathbf{E} \left[\sum_{l = -L+1}^{L - 1} \beta_l \rvw_{0, l}\right]^2 - \frac{1}{2} \log(1 - 2\theta \mathcal{E}_s) \quad \mbox{a.s.}
\end{eqnarray}
In both limits above, $\{\rvw_{0, l}\}_{l = -L+1}^{L - 1}$ are induced by an infinite sequence of inputs, $\{\rvx_k\}_{k = -\infty}^\infty$.

So the GMI is
\begin{eqnarray}
\label{eqn:i-gmi-super-nyquist}
I_\mathrm{GMI} = \sup_{\underline{\beta}, \theta < 0} \left\{\theta \mathbf{E} \left[\sum_{l = -L+1}^{L - 1} \beta_l \rvw_{0, l} - \rvx_0\right]^2 - \frac{\theta}{1 - 2\theta \mathcal{E}_s} \mathbf{E} \left[\sum_{l = -L+1}^{L - 1} \beta_l \rvw_{0, l}\right]^2 + \frac{1}{2} \log(1 - 2\theta \mathcal{E}_s) \right\},
\end{eqnarray}
and we have the following result, whose derivation is given in Supplementary Material \ref{appendix:proof-gmi-super-nyquist}.
\begin{prop}
\label{prop:gmi-super-nyquist}
The GMI with super-Nyquist output sampling is
\begin{eqnarray}
\label{eqn:gmi-super-nyquist}
I_\mathrm{GMI} = \frac{1}{2}\log\left(1 + \frac{\Delta}{1 - \Delta}\right),
\end{eqnarray}
where $\Delta = \left(\underline{b}^T \mathbf{\Omega}^{-1} \underline{b}\right)/{\mathcal{E}_s}$, $\mathbf{\Omega}$ is a $(2L-1) \times (2L-1)$ matrix with its $(u, l)$-element being $\mathbf{E}[\rvw_{0, u} \rvw_{0, l}]$, and $\underline{b}$ is a $(2L-1)$-dimensional vector with its $l$-element being $\mathbf{E}[\rvx_0 \rvw_{0, l}]$, $u, l = -L+1, \ldots, L-1$. To achieve the GMI in (\ref{eqn:gmi-super-nyquist}), the optimal weighting coefficients are
\begin{eqnarray}
\underline{\beta} = \frac{\mathcal{E}_s}{\underline{b}^T \mathbf{\Omega}^{-1} \underline{b}} \mathbf{\Omega}^{-1} \underline{b}.
\end{eqnarray}
\end{prop}

We notice that the GMI in (\ref{eqn:gmi-super-nyquist}) is a natural extension of that in Proposition \ref{prop:gmi-real} for the Nyquist-sampling case, and we can also define the effective SNR by $\mathrm{SNR}_e = \Delta/(1 - \Delta)$.

\subsection{Binary Symmetric Quantization: Sinc Pulse Function}

We examine binary symmetric quantization in which $w = \mathrm{sgn}(y)$. For this purpose, we need to evaluate $\mathbf{\Omega}$ and $\underline{b}$. For each $l$,
\begin{eqnarray}
\rvy_{0, l} = \frac{1}{\sqrt{2W}} \sum_{k = -\infty}^\infty \rvx_k g\left(\frac{l}{2WL} - \frac{k}{2W}\right) + \rvz\left(\frac{l}{2WL}\right).
\end{eqnarray}

Utilizing (\ref{eqn:x-sgny}) and noting that $\{\rvx_k\}$ are i.i.d., we have
\begin{eqnarray}
\label{eqn:b-vector}
b_l &=& \mathbf{E}[\rvx_0 \mathrm{sgn}(\rvy_{0, l})]\nonumber\\
&=& \frac{\mathcal{E}_s g(l/(2WL))}{\sqrt{\pi \left[(\mathcal{E}_s/2)\sum_{k = -\infty}^\infty g^2(l/(2WL) - k/(2W)) + \sigma^2 W/2\right]}},
\end{eqnarray}
for $l = -L+1, \ldots, L - 1$.

The undistorted received signal samples, $\rvy_{0, u}$ and $\rvy_{0, l}$, are jointly zero-mean Gaussian. We can further evaluate their correlation as
\begin{eqnarray}
\label{eqn:r-matrix}
&&r_{u, l} = \frac{\mathbf{E} [\rvy_{0, u} \rvy_{0, l}]}{\sqrt{\mathrm{var}[\rvy_{0, u}]}\cdot \sqrt{\mathrm{var}[\rvy_{0, l}]}}\nonumber\\
&=& \frac{\frac{\mathcal{E}_s}{2W}\sum_{k = -\infty}^\infty g\left(l/(2WL) - k/(2W)\right) g\left(u/(2WL) - k/(2W)\right) + \frac{\sigma^2}{2} \mathrm{sinc}\left((l - u)/L\right)}{\sqrt{\frac{\mathcal{E}_s}{2W} \sum_{k = -\infty}^\infty g^2(l/(2WL) - k/(2W)) + \frac{\sigma^2}{2}} \sqrt{\frac{\mathcal{E}_s}{2W} \sum_{k = -\infty}^\infty g^2(u/(2WL) - k/(2W)) + \frac{\sigma^2}{2}}}.\nonumber
\end{eqnarray}
Consequently, the correlation between the hard-limited samples is \cite{vanvleck66:pieee}
\begin{eqnarray}
\Omega_{u, l} = \mathbf{E}[\rvw_{0, u} \rvw_{0, l}] = \frac{2}{\pi} \arcsin r_{u, l}.
\end{eqnarray}

Now in this subsection we focus on the sinc pulse function, $g(t) = \sqrt{2W} \mathrm{sinc}(2Wt)$. For this $g(\cdot)$, through (\ref{eqn:b-vector}) and (\ref{eqn:r-matrix}) we have
\begin{eqnarray}
b_l &=& \frac{2\mathcal{E}_s}{\sqrt{\pi \sigma^2}} \frac{\mathrm{sinc}(l/L)}{\sqrt{(2\mathcal{E}_s/\sigma^2) \Xi(l, l) + 1}},\\
r_{u, l} &=& \frac{(2 \mathcal{E}_s/\sigma^2) \Xi(l, u) + \mathrm{sinc}\left(\frac{l - u}{L}\right)}{\sqrt{(2 \mathcal{E}_s/\sigma^2)\Xi(l, l) + 1} \sqrt{(2 \mathcal{E}_s/\sigma^2) \Xi(u, u) + 1}},
\end{eqnarray}
where $\Xi(l, u) = \sum_{k = -\infty}^\infty \mathrm{sinc}\left(l/L - k\right) \mathrm{sinc}\left(u/L - k\right)$, which can be further evaluated as $\Xi(l, u) = \mathrm{sinc}((l - u)/L)$, for all $l, u = -L+1, \ldots, L - 1$. So
\begin{eqnarray}
b_l = \sqrt{\frac{2\mathcal{E}_s}{\pi}} \sqrt{\frac{2\mathcal{E}_s/\sigma^2}{2\mathcal{E}_s/\sigma^2 + 1}} \mathrm{sinc}\left(l/L\right), \quad\mbox{and}\; r_{u, l} = \mathrm{sinc}\left(\frac{l - u}{L}\right).
\end{eqnarray}

When $L = 1$, {\it i.e.}, Nyquist sampling, we can easily verify that $\Delta = \frac{2}{\pi} \frac{\mathcal{E}_s}{\mathcal{E}_s + \sigma^2/2}$, thus revisiting the result in Section \ref{sec:1-bit}.

From the above, we can find the following behavior of the GMI, in which we denote $\mathrm{SNR} = \frac{\mathcal{E}_s}{\sigma^2/2}$, $\underline{b}_0 = \left[\mathrm{sinc}\left(l/L\right)\right]_{l = -L+1, \ldots, L - 1}$, and $\mathbf{\Omega}_0 = \left[\arcsin \mathrm{sinc}\left((l - u)/L\right)\right]_{l, u = -L+1, \ldots, L - 1}$.
\begin{eqnarray}
\Delta = \frac{\mathrm{SNR}}{\mathrm{SNR} + 1} \underline{b}_0^T \mathbf{\Omega}_0^{-1} \underline{b}_0,\quad\mbox{and}\; \mathrm{SNR}_e = \frac{\underline{b}_0^T \mathbf{\Omega}_0^{-1} \underline{b}_0 \cdot \mathrm{SNR}}{(1 - \underline{b}_0^T \mathbf{\Omega}_0^{-1} \underline{b}_0)\cdot \mathrm{SNR} + 1}.
\end{eqnarray}
\begin{itemize}
\item High-SNR regime: As $\mathrm{SNR} \rightarrow \infty$,
\begin{eqnarray}
I_\mathrm{GMI} = \frac{1}{2} \log \left(\frac{1}{1 - \underline{b}_0^T \mathbf{\Omega}_0^{-1} \underline{b}_0}\right) + o(1).
\end{eqnarray}
\item Low-SNR regime: As $\mathrm{SNR} \rightarrow 0$,
\begin{eqnarray}
I_\mathrm{GMI} = \frac{\underline{b}_0^T \mathbf{\Omega}_0^{-1} \underline{b}_0}{2} \mathrm{SNR} + o(\mathrm{SNR}).
\end{eqnarray}
\end{itemize}

In Table \ref{tab:super-nyquist}, we present the numerical results for the asymptotic behavior of the GMI, for different values of $L$.
\begin{table}[ht]
\vspace{0.1in}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
  \hline

  $L$ & 1 & 2 & 4 & 8 & 16 & 32 & $\infty$ \\
  \hline
  $\underline{b}_0^T \mathbf{\Omega}_0^{-1} \underline{b}_0$ & $2/\pi$ & 0.7173 & 0.7591 & 0.7734 & 0.7783 & 0.7801 & 0.7815 \\
  \hline
  $\lim_{\mathrm{SNR} \rightarrow \infty} I_\mathrm{GMI}$ (bits/c.u.) & 0.7302 & 0.9114 & 1.0268 & 1.0710 & 1.0867 & 1.0926 & 1.0970 \\
  \hline
  $\lim_{\mathrm{SNR} \rightarrow 0} I_\mathrm{GMI}/\mathrm{SNR}$ & 0.3183 & 0.3587 & 0.3796 & 0.3867 & 0.3892 & 0.3901 & 0.3907 \\
  \hline
\end{tabular}
\vspace{0.1in}
\caption{Table of performance for super-Nyquist output sampling with binary symmetric quantization and sinc pulse function.}
\label{tab:super-nyquist}
\end{table}
From the numerical results, we see that super-Nyquist sampling yields noticeable benefit for the GMI. In the low-SNR regime, sampling at twice the Nyquist rate attains $\lim_{\mathrm{SNR} \rightarrow 0} I_\mathrm{GMI}/\mathrm{SNR} = 0.3587$, which is slightly smaller than the lower bound $0.3732$ which has been obtained in \cite{koch10:arxiv}. In the high-SNR regime, we further observe that for $L \geq 4$ the GMI exceeds $1$ bit/c.u.! Intuitively, this is due to the fact that the diversity yielded by super-Nyquist sampling is capable of exploiting the abundant information carried by the Gaussian codebook ensemble.

To further consolidate our above analysis, in Figure \ref{fig:super-nyquist} we plot the GMI achieved for different values of $L$. We can clearly observe the rate gain by increasing the sampling rate. For comparison, we also plot the AWGN capacity without distortion and the capacity under binary symmetric quantization and with Nyquist sampling \cite{singh09:com}. We notice that, as $L$ increases, on one hand, the performance gap between the GMI and the capacity tends to diminish for SNR smaller than 0 dB; on the other hand, the GMI even outperforms the capacity at high SNR.
\begin{figure}[ht]
\centering
\includegraphics[width=3.5in]{./figs/super_nyquist_curve.eps}
\caption{The GMI achieved by super-Nyquist sampling with binary symmetric quantization and sinc pulse function, for $L = 1, 2, 4, 8, 16$.}
\label{fig:super-nyquist}
\end{figure}

\subsection{Binary Symmetric Quantization: Pulse Function Optimization at Low SNR}

We have already seen in the previous subsection that super-Nyquist sampling yields noticeable benefit. In this subsection, we illustrate that we can even realize additional benefit through optimizing the pulse function $g(\cdot)$.

With sampling factor $L$, we restrict the pulse function to take the following form
\begin{eqnarray}
g(t) = \sum_{v = -L+1}^{L-1} \gamma_v \sqrt{2W} \mathrm{sinc}(2Wt - v/L);
\end{eqnarray}
that is, a superposition of $2L - 1$ (time-shifted) sinc pulses. The weighting parameters $\{\gamma_v\}_{v = -L+1}^{L-1}$ are such that the energy of $g(t)$ is unity, {\it i.e.},
\begin{eqnarray}
\int_{-\infty}^\infty g^2(t) dt = \sum_{v = -L+1}^{L-1} \sum_{v^\prime = -L+1}^{L-1} \gamma_v \gamma_{v^\prime} \mathrm{sinc}\left(\frac{v - v^\prime}{L}\right) = 1,
\end{eqnarray}
which may be rewritten in matrix form as $\underline{\gamma}^T \mathbf{\Theta} \underline{\gamma} = 1$, where $\mathbf{\Theta} = \left[\mathrm{sinc}\left((l - u)/L\right)\right]_{l, u = -L+1, \ldots, L - 1}$. If we let $\gamma_0 = 1$ and $\gamma_{v \neq 0} = 0$, we obtain the sinc pulse function.

Through the general formulas (\ref{eqn:b-vector}) and (\ref{eqn:r-matrix}), we have, after some algebraic manipulation,
\begin{eqnarray}
b_l &=& \sqrt{\frac{2\mathcal{E}_s}{\pi}} \sqrt{\frac{2\mathcal{E}_s/\sigma^2}{2\mathcal{E}_s/\sigma^2 + 1}} \sum_{v = -L+1}^{L-1} \gamma_v \mathrm{sinc}\left(\frac{l - v}{L}\right),\\
r_{u, l} &=& \frac{(2\mathcal{E}_s/\sigma^2) \sum_{a = -L+1}^{L-1}\sum_{b = -L+1}^{L-1} \gamma_a \gamma_b \mathrm{sinc}\left(\frac{l - u - a + b}{L}\right) + \mathrm{sinc}\left(\frac{l - u}{L}\right)}{2\mathcal{E}_s/\sigma^2 + 1}.
\end{eqnarray}

To illustrate the benefit of optimizing the pulse function, we focus on the low-SNR regime, where $\mathrm{SNR} = \frac{\mathcal{E}_s}{\sigma^2/2}$ approaches toward zero. We thus have
\begin{eqnarray}
\sqrt{\frac{\pi}{2\mathcal{E}_s}} \frac{b_l}{\sqrt{\mathrm{SNR}}} \rightarrow \sum_{v = -L+1}^{L-1} \gamma_v \mathrm{sinc}\left(\frac{l - v}{L}\right), \quad\mbox{and}\;\; r_{u, l} &\rightarrow& \mathrm{sinc}\left(\frac{l - u}{L}\right).
\end{eqnarray}
Subsequently, the value of $\Delta$ and $\mathrm{SNR}_e$ in Proposition \ref{prop:gmi-super-nyquist} behaves like
\begin{eqnarray}
\lim_{\mathrm{SNR} \rightarrow 0}\frac{\mathrm{SNR}_e}{\mathrm{SNR}} = \lim_{\mathrm{SNR} \rightarrow 0} \frac{\Delta}{\mathrm{SNR}} = \underline{\gamma}^T \mathbf{\Theta} \mathbf{\Omega}_0^{-1} \mathbf{\Theta} \underline{\gamma},
\end{eqnarray}
where $\mathbf{\Theta} = \left[\mathrm{sinc}\left((l - u)/L\right)\right]_{l, u = -L+1, \ldots, L - 1}$ and $\mathbf{\Omega}_0 = \left[\arcsin \mathrm{sinc}\left((l - u)/L\right)\right]_{l, u = -L+1, \ldots, L - 1}$ have been defined previously. Keeping in mind the unit-energy constraint on $g(t)$, the following optimization problem is immediate,
\begin{eqnarray}
\max_{\underline{\gamma}} \underline{\gamma}^T \mathbf{\Theta} \mathbf{\Omega}_0^{-1} \mathbf{\Theta} \underline{\gamma}, \quad \mbox{s.t.}\;\; \underline{\gamma}^T \mathbf{\Theta} \underline{\gamma} = 1.
\end{eqnarray}
By noting that $\mathbf{\Theta}$ is a positive-definite matrix, we can introduce the transform $\tilde{\underline{\gamma}} = \mathbf{\Theta}^{1/2} \underline{\gamma}$, and rewrite the optimization problem as
\begin{eqnarray}
\max_{\underline{\gamma}} \frac{\tilde{\underline{\gamma}}^T \mathbf{\Theta}^{1/2} \mathbf{\Omega}_0^{-1} \mathbf{\Theta}^{1/2} \tilde{\underline{\gamma}}}{\tilde{\underline{\gamma}}^T \tilde{\underline{\gamma}}},
\end{eqnarray}
for which the maximum value is the largest eigenvalue of $\mathbf{\Theta}^{1/2} \mathbf{\Omega}_0^{-1} \mathbf{\Theta}^{1/2}$, and the optimal $\tilde{\underline{\gamma}}$ is the unit-norm eigenvector corresponding to the largest eigenvalue.

In Table \ref{tab:super-nyquist-opt-pulse}, we present the numerical results for the low-SNR asymptotic behavior of the GMI, with the optimal choice of $\underline{\gamma}$, for different values of $L$.
\begin{table}[ht]
\vspace{0.1in}
\begin{tabular}{|l|l|l|l|l|l|l|}
  \hline

  $L$ & 2 & 4 & 8 & 16 & 32 & $\infty$ \\
  \hline
  $\lim_{\mathrm{SNR} \rightarrow 0} I_\mathrm{GMI}/\mathrm{SNR}$ & 0.3731 & 0.3923 & 0.3971 & 0.3984 & 0.3987 & 0.3988 \\
  \hline
\end{tabular}
\vspace{0.1in}
\caption{Table of performance for super-Nyquist output sampling with binary symmetric quantization and optimized pulse function.}
\label{tab:super-nyquist-opt-pulse}
\end{table}
Compared with Table \ref{tab:super-nyquist}, we notice that optimizing the pulse function leads to a noticeable additional improvement on the GMI. In particular, for $L = 2$ our approach yields $\lim_{\mathrm{SNR} \rightarrow 0} I_\mathrm{GMI}/\mathrm{SNR} = 0.3731$, which almost coincides with the result in \cite{koch10:arxiv}, $0.3732$.\footnote{Since both our result and that in \cite{koch10:arxiv} are analytical, we have compared their values in fine precision and found that they are indeed different.}

\section{Conclusions}
\label{sec:conclusions}

With the surging quest for energy-efficient communication solutions, transceivers with deliberately engineered distortions have attracted much attention in system design. These distortions, such as transmit-side clipping and low-precision receive-side quantization, may significantly alleviate power consumption and hardware cost. It is thus imperative for communication engineers to develop a systematic understanding of the impact of these distortions, so as to assess the resulting system performance, and to guide the design of distortion mechanisms. In this paper, we make an initial attempt at this goal, developing a general analytical framework for evaluating the achievable information rates using the measure of GMI, and illustrating the application of this framework by examining several representative transceiver distortion models. We hope that both the framework and the applications presented in this paper will be useful for deepening our understanding in this area.

Admittedly, the approach taken in this paper, namely evaluating the GMI for Gaussian codebook ensemble and nearest-neighbor decoding, is inherently suboptimal for general transceiver distortion models. Nevertheless, as illustrated throughout this paper, the general analytical framework built upon such an approach is convenient for performance evaluation and instrumental for system design. In many practically important scenarios, for example the low/moderate-SNR regime, this approach leads to near-optimal performance. Furthermore, as suggested by our analysis of super-Nyquist sampling, we can substantially alleviate the performance loss by sampling the channel output at rates higher than the Nyquist rate.

A number of interesting problems remain unsolved within the scope of this paper. These include, among others: answering whether the GMI coincides with the channel capacity for multi-bit output quantization in the low-SNR limit; identifying more effective ways of processing the samples in super-Nyquist sampled channels; characterizing the ultimate performance limit of super-Nyquist sampling. Beyond the scope of this paper, one can readily see a whole agenda of research on communication with nonlinear transceiver distortion, including timing recovery, channel estimation, equalization, transmission under multipath fading, and multiantenna/multiuser aspects.

\bibliographystyle{ieee}

redpajama_set_name: RedPajamaArXiv


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A young man spoken of in stories from the faraway land of Eos. Ignis was born into the Scientia family, retainers of the royal family of Lucis, and as such was raised to be Prince Noctis's companion since childhood. Even after joining the Crownsguard upon finishing high school he looked after the prince's needs on a daily basis, and is expected to continue supporting him after the latter is eventually crowned king. Ignis is a talented cook, and enjoys coming up with many new and original recipes.
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Is there something you're holding onto? If you've any worries, then tell me.


text:
package org.apache.flink.streaming.runtime.operators;

import org.apache.flink.api.common.typeinfo.TypeInformation;
import org.apache.flink.runtime.checkpoint.OperatorSubtaskState;
import org.apache.flink.streaming.runtime.streamrecord.StreamRecord;
import org.apache.flink.streaming.util.AbstractStreamOperatorTestHarness;
import org.apache.flink.streaming.util.OneInputStreamOperatorTestHarness;
import org.apache.flink.util.TestLoggerExtension;

import org.junit.jupiter.api.Test;
import org.junit.jupiter.api.extension.ExtendWith;

/** Test base for {@link GenericWriteAheadSink}. */
@ExtendWith(TestLoggerExtension.class)
public abstract class WriteAheadSinkTestBase<IN, S extends GenericWriteAheadSink<IN>> {

    protected abstract S createSink() throws Exception;

    protected abstract TypeInformation<IN> createTypeInfo();

    protected abstract IN generateValue(int counter, int checkpointID);

    protected abstract void verifyResultsIdealCircumstances(S sink) throws Exception;

    protected abstract void verifyResultsDataPersistenceUponMissedNotify(S sink) throws Exception;

    protected abstract void verifyResultsDataDiscardingUponRestore(S sink) throws Exception;

    protected abstract void verifyResultsWhenReScaling(
            S sink, int startElementCounter, int endElementCounter) throws Exception;

    private final int maxParallelism = 10;

    @Test
    void testIdealCircumstances() throws Exception {
        S sink = createSink();

        OneInputStreamOperatorTestHarness<IN, IN> testHarness =
                new OneInputStreamOperatorTestHarness<>(sink);

        testHarness.open();

        int elementCounter = 1;
        int snapshotCount = 0;

        for (int x = 0; x < 20; x++) {
            testHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 0)));
            elementCounter++;
        }

        testHarness.snapshot(snapshotCount++, 0);
        testHarness.notifyOfCompletedCheckpoint(snapshotCount - 1);

        for (int x = 0; x < 20; x++) {
            testHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 1)));
            elementCounter++;
        }

        testHarness.snapshot(snapshotCount++, 0);
        testHarness.notifyOfCompletedCheckpoint(snapshotCount - 1);

        for (int x = 0; x < 20; x++) {
            testHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 2)));
            elementCounter++;
        }

        testHarness.snapshot(snapshotCount++, 0);
        testHarness.notifyOfCompletedCheckpoint(snapshotCount - 1);

        verifyResultsIdealCircumstances(sink);
    }

    @Test
    void testDataPersistenceUponMissedNotify() throws Exception {
        S sink = createSink();

        OneInputStreamOperatorTestHarness<IN, IN> testHarness =
                new OneInputStreamOperatorTestHarness<>(sink);

        testHarness.open();

        int elementCounter = 1;
        int snapshotCount = 0;

        for (int x = 0; x < 20; x++) {
            testHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 0)));
            elementCounter++;
        }

        testHarness.snapshot(snapshotCount++, 0);
        testHarness.notifyOfCompletedCheckpoint(snapshotCount - 1);

        for (int x = 0; x < 20; x++) {
            testHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 1)));
            elementCounter++;
        }

        testHarness.snapshot(snapshotCount++, 0);

        for (int x = 0; x < 20; x++) {
            testHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 2)));
            elementCounter++;
        }

        testHarness.snapshot(snapshotCount++, 0);
        testHarness.notifyOfCompletedCheckpoint(snapshotCount - 1);

        verifyResultsDataPersistenceUponMissedNotify(sink);
    }

    @Test
    void testDataDiscardingUponRestore() throws Exception {
        S sink = createSink();

        OneInputStreamOperatorTestHarness<IN, IN> testHarness =
                new OneInputStreamOperatorTestHarness<>(sink);

        testHarness.open();

        int elementCounter = 1;
        int snapshotCount = 0;

        for (int x = 0; x < 20; x++) {
            testHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 0)));
            elementCounter++;
        }

        OperatorSubtaskState latestSnapshot = testHarness.snapshot(snapshotCount++, 0);
        testHarness.notifyOfCompletedCheckpoint(snapshotCount - 1);

        for (int x = 0; x < 20; x++) {
            testHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 1)));
            elementCounter++;
        }

        testHarness.close();

        sink = createSink();

        testHarness = new OneInputStreamOperatorTestHarness<>(sink);

        testHarness.setup();
        testHarness.initializeState(latestSnapshot);
        testHarness.open();

        for (int x = 0; x < 20; x++) {
            testHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 2)));
            elementCounter++;
        }

        testHarness.snapshot(snapshotCount++, 0);
        testHarness.notifyOfCompletedCheckpoint(snapshotCount - 1);

        verifyResultsDataDiscardingUponRestore(sink);
    }

    @Test
    void testScalingDown() throws Exception {
        S sink1 = createSink();
        OneInputStreamOperatorTestHarness<IN, IN> testHarness1 =
                new OneInputStreamOperatorTestHarness<>(sink1, maxParallelism, 2, 0);
        testHarness1.open();

        S sink2 = createSink();
        OneInputStreamOperatorTestHarness<IN, IN> testHarness2 =
                new OneInputStreamOperatorTestHarness<>(sink2, maxParallelism, 2, 1);
        testHarness2.open();

        int elementCounter = 1;
        int snapshotCount = 0;

        for (int x = 0; x < 10; x++) {
            testHarness1.processElement(new StreamRecord<>(generateValue(elementCounter, 0)));
            elementCounter++;
        }

        for (int x = 0; x < 11; x++) {
            testHarness2.processElement(new StreamRecord<>(generateValue(elementCounter, 0)));
            elementCounter++;
        }

        // snapshot at checkpoint 0 for testHarness1 and testHarness 2
        OperatorSubtaskState snapshot1 = testHarness1.snapshot(snapshotCount, 0);
        OperatorSubtaskState snapshot2 = testHarness2.snapshot(snapshotCount, 0);

        // merge the two partial states
        OperatorSubtaskState mergedSnapshot =
                AbstractStreamOperatorTestHarness.repackageState(snapshot1, snapshot2);

        testHarness1.close();
        testHarness2.close();

        // and create a third instance that operates alone but
        // has the merged state of the previous 2 instances

        OperatorSubtaskState initState =
                AbstractStreamOperatorTestHarness.repartitionOperatorState(
                        mergedSnapshot, maxParallelism, 2, 1, 0);

        S sink3 = createSink();
        OneInputStreamOperatorTestHarness<IN, IN> mergedTestHarness =
                new OneInputStreamOperatorTestHarness<>(sink3, maxParallelism, 1, 0);

        mergedTestHarness.setup();
        mergedTestHarness.initializeState(initState);
        mergedTestHarness.open();

        for (int x = 0; x < 12; x++) {
            mergedTestHarness.processElement(new StreamRecord<>(generateValue(elementCounter, 0)));
            elementCounter++;
        }

        snapshotCount++;
        mergedTestHarness.snapshot(snapshotCount, 1);
        mergedTestHarness.notifyOfCompletedCheckpoint(snapshotCount);

        verifyResultsWhenReScaling(sink3, 1, 33);
        mergedTestHarness.close();
    }

    @Test
    void testScalingUp() throws Exception {

        S sink1 = createSink();
        OneInputStreamOperatorTestHarness<IN, IN> testHarness1 =
                new OneInputStreamOperatorTestHarness<>(sink1, maxParallelism, 1, 0);

        int elementCounter = 1;
        int snapshotCount = 0;

        testHarness1.open();

        // put two more checkpoints as pending

        for (int x = 0; x < 10; x++) {
            testHarness1.processElement(new StreamRecord<>(generateValue(elementCounter, 0)));
            elementCounter++;
        }
        testHarness1.snapshot(++snapshotCount, 0);

        for (int x = 0; x < 11; x++) {
            testHarness1.processElement(new StreamRecord<>(generateValue(elementCounter, 0)));
            elementCounter++;
        }

        // this will be the state that will be split between the two new operators
        OperatorSubtaskState snapshot = testHarness1.snapshot(++snapshotCount, 0);

        testHarness1.close();

        // verify no elements are in the sink
        verifyResultsWhenReScaling(sink1, 0, -1);

        // we will create two operator instances, testHarness2 and testHarness3,
        // that will share the state of testHarness1

        OperatorSubtaskState initState1 =
                AbstractStreamOperatorTestHarness.repartitionOperatorState(
                        snapshot, maxParallelism, 1, 2, 0);

        OperatorSubtaskState initState2 =
                AbstractStreamOperatorTestHarness.repartitionOperatorState(
                        snapshot, maxParallelism, 1, 2, 1);

        ++snapshotCount;

        S sink2 = createSink();
        OneInputStreamOperatorTestHarness<IN, IN> testHarness2 =
                new OneInputStreamOperatorTestHarness<>(sink2, maxParallelism, 2, 0);

        testHarness2.setup();
        testHarness2.initializeState(initState1);
        testHarness2.open();

        testHarness2.notifyOfCompletedCheckpoint(snapshotCount);

        verifyResultsWhenReScaling(sink2, 1, 10);

        S sink3 = createSink();
        OneInputStreamOperatorTestHarness<IN, IN> testHarness3 =
                new OneInputStreamOperatorTestHarness<>(sink3, maxParallelism, 2, 1);

        testHarness3.setup();
        testHarness3.initializeState(initState2);
        testHarness3.open();

        // add some more elements to verify that everything functions normally from now on...

        for (int x = 0; x < 10; x++) {
            testHarness3.processElement(new StreamRecord<>(generateValue(elementCounter, 0)));
            elementCounter++;
        }

        testHarness3.snapshot(snapshotCount, 1);
        testHarness3.notifyOfCompletedCheckpoint(snapshotCount);

        verifyResultsWhenReScaling(sink3, 11, 31);

        testHarness2.close();
        testHarness3.close();
    }
}

redpajama_set_name: RedPajamaGithub


text:
ACCEPTED

#### According to
Index Fungorum

#### Published in
Bulletino dell'orto Botanico della R. Universitá di Napoli 6: 71 (1921)

#### Original name
Cladosporium densum Sacc.

### Remarks
null


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Click the "Apply for Permit" button to download a copy of the application to your device.
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Please include in your application a detailed location of the burn location (Physical address for entry gate). Please fill in all the Permit conditions you can, including the type and number of fire appliances/machinery and size of fire breaks in accordance with the Fire Break order. The remaining conditions will be finalised by your fire districts Bush Fire Control Officer.
Step 3:- Click the Submit Button and your application will be sent via email for processing by our staff.
Step 4:- Once the permit is received and processed. A Bush Fire Control Officer will fill in any additional conditions and issue a Permit number and send you an email copy of the signed Permit .
If the details of the permit need to be clarified your local Bush Fire Control Officer will contact you to get further details or book in a time to inspect your burn and approve the permit.
Step 5:- Read and follow all conditions on the Permit. Call DFES Communications Centre on 1800 198 140 to log your burn. Your burn will be placed on the www.emergency.wa.gov.au website so the public will be aware of the burn off.
The burning of garden refuse or rubbish on the ground or in an incinerator that would otherwise be permitted under Section 24F is prohibited on all land under 2000m2 (square metres) in size during the Restricted Burning Time, without a Fire Permit issued under Regulation 15 of the Bush Fires Act 1954.
Permits are available for the Restricted Period by contacting the Fire Control Officer in your area from the list below.


text:
Home > Projects > Business and Human Rights
THE ISSUE Coal Mining and Adivasi Rights For decades, India's Adivasis have borne the brunt of development-induced displacement, including through state-run coal mines. A range of protective laws have not prevented indigenous communities in India from having their lands taken, their livelihoods destroyed, and their rights trampled on as a result of business activities. Coal India Limited – the world's largest coal miner – and its subsidiaries, central government ministries and state government authorities have failed to ensure meaningful consultation with Adivasi communities on land acquisition, rehabilitation and resettlement and the environmental impacts of mines. Thus, seriously affecting the lives and livelihoods of Adivasi communities. Adivasis, who constitute about 8% of India's population, rely on their lands and forests for their livelihoods. They have been frequently displaced from their lands by laws such as the Coal Bearing Areas (Acquisition and Development) Act, which does not require authorities to consult affected communities or seek the free, prior and informed consent of indigenous peoples, as stipulated under international law and standards. Laws such as the Scheduled Castes and Scheduled Tribes (Prevention of Atrocities) Act, which criminalize the dispossession of Adivasi land without consent, are rarely enforced. AMNESTY IN ACTION We work with Adivasi communities to empower them to claim their rights under Indian and international law, and to ensure that their rights are respected by coal mining companies and protected by government authorities.
India Regions
Chhattisgarh | Jharkhand | Odisha
#ForIndigenousRights
On 14 June 2017, over 80 Adivasis from four villages in Raigarh district came together at the district's Scheduled Castes and Scheduled Tribes Welfare (Special) Police Station to lodge complaints against people they said were agents of two companies
India: Devastating Supreme Court ruling could render over a million indigenous people homeless
This ruling is a body blow to Adivasi rights in India. It would be unconscionable for Adivasi people to be evicted from their homes and lands merely because their claims were rejected in a process that is known to be severely flawed.
85 Adivasi villagers in Raigarh called upon the Chhattisgarh police to conduct a prompt, independent and impartial investigation
85 Adivasi villagers in Raigarh, Chhattisgarh, called upon the Chhattisgarh police to conduct a prompt, independent and impartial investigation into allegations that they had been unlawfully dispossessed of their lands by people acting as agents for two private companies.
Government Circular Attacks Dalit Identity In India
The Ministry of Information and Broadcasting's circular to television news channels to refrain from using the term 'Dalit' and use the term 'Scheduled Caste' instead, will impede the Dalit rights movement from asserting itself against caste-based oppression
क्षेत्रीय कहानियाँ: क्या आपको पता है कि रायगढ़ में क्या हो रहा है?
मैंने छत्तीसगढ़ के छोटे से जिले, रायगढ़ में पिछले कुछ हफ्ते बिताए। रायगढ़ ऐतिहासिक रूप से विभिन्न आदिवासी समुदायों से
Ordinary Dreams, Extraordinary Circumstances
Manohar Chauhan, Senior Campaigner, interviewed Rebati Chhatria, the woman who's standing up for Adivasi rights.
FIELD STORIES: Do You Know What's Happening In Raigarh?
Our campaigner recently visted Raigarh as part of an evaluation of our campaign work in the district and shares his experience on field.
The Cost of Coal
Amnesty International India's Submission on the draft Compensatory Afforestation Fund Rules, 2018


Check out this mixtape, and also check out http://www.dvrylmuse.com for more photography and art!
Original version of Jay-Z's Dip Set diss from "Kingdom Come". This instrumental later on appeared on Fabolous's "From Nothing To Something" album.


President's Secretariat Notification No 1-Pres/50 dated 26 Jan 50.
For acts of gallantry in the presence of the enemy on land, at sea or in the air.If any recipient of the Chakra shall again perform such an act of bravery as would have made him or her eligible to receive the Chakra, such further act of bravery shall be recorded by a Bar to be attached to the riband by which the Chakra is suspended, and for every such additional act of bravery, an additional bar shall be added, and any such bar or bars may also be awarded posthumously. For every bar awarded a replica of the Chakra in miniature shall be added to the riband when worn alone. The decoration may be awarded posthumously.
Monetary allowance : Rs 5,000/- pm and each bar to the decoration Rs 5,000/- pm to all recipients.
Medal Circular in shape and is made of standard silver, and is embossed on the obverse a five pointed heraldic star with the points of the star just touching the rim. Medal shall be one and three eighth inches in diameter. The State emblem (including motto) is embossed in the centre-piece which is domed. The star is polished and the centre piece is in gold gilt. On the reverse it will have embossed Maha Vir Chakra both in Hindi and English with two lotus flowers between Hindi and English wording. The fitting is swivel mounting.
Riband The Riband is of a half white and half orange colour.


text:
La ligne 2 du tramway de Nantes est une ligne du réseau de transports en commun de Nantes Métropole exploitée par la Semitan.

Lancée le  et inaugurée le , la ligne 2 permet, sur un axe nord-sud, de relier les villes d'Orvault, Nantes et Rezé.

La ligne a assuré le déplacement de 18,7 millions de voyageurs en 2009, soit une moyenne de  par jour. Pour la période 2014/2015,  voyages par jour ont été effectués sur la ligne.

Histoire

Après le succès de la ligne 1, Nantes Métropole décide, vers la fin des années 1980, de se doter d'une seconde ligne nord-sud, afin notamment de reconquérir le centre-ville en passant par le cours des 50-Otages.

Les travaux de cette nouvelle ligne ont également permis de reconfigurer le cours des 50-Otages, passant d'un large boulevard dévolu à la circulation automobile à une voie en zone 30 dont une partie serait prise par la ligne 2. Au total,  d'espace public ont été reconfigurés.

Les premières rames circulent à partir du 7 septembre 1992 : elles relient Trocardière à Commerce, puis 50 Otages à partir du 19 décembre 1992, après la fin des derniers travaux. Pour l'anecdote, les rames ont arboré de grands nœuds verts à l'avant lors du lancement de la ligne. Cette nouvelle ligne est inaugurée le 26 septembre 1992 par Jean-Louis Bianco, alors Ministre des Transports.

Après de longs travaux, un nouveau tronçon reliant 50 Otages et École Centrale ─ Sup de Co (actuel École Centrale ─ Audencia) en 10 minutes est inauguré le 3 septembre 1993 puis mis en service le 6 septembre.

En 1994, le tronçon École Centrale ─ Sup de Co ↔ Orvault ─ Grand Val ouvre progressivement, avec un premier prolongement jusqu'à Bourgeonnière ouvert le 14 mars, puis jusqu'à Orvault et le nouveau terminus le 29 août. Ce nouveau tronçon est inauguré le 3 septembre 1994.

Le 29 août 2005, la ligne 2 est prolongée au sud jusqu'au nouveau terminus Neustrie afin de desservir la commune de Bouguenais. Ce nouveau tronçon est inauguré le 3 septembre 2005.

Le 27 août 2007, la ligne 2 étant l'une des plus fréquentées de France et donc fréquemment saturée aux heures de pointe, les lignes 2 et 3 sont interverties au sud-Loire afin de la raccourcir : la ligne 3 reprend l'itinéraire de la ligne 2 jusqu'à Neustrie, et cette dernière est limitée au nouveau terminus Gare de Pont-Rousseau, initialement prévu pour la ligne 3.

Infrastructure

La ligne

La ligne 2 du tramway de Nantes emprunte les sections suivantes :
 Gare de Pont Rousseau ↔ Commerce, ouverte le 7 septembre 1992, à l'occasion de la mise en service de la ligne (la station Gare de Pont Rousseau a été ouverte le 27 août 2007 lors de l'inversion des lignes 2 et 3 au sud-Loire) ;
 Commerce ↔ 50 Otages, ouverte le 19 décembre 1992, à l'occasion du premier prolongement de la ligne ;
 50 Otages ↔ École Centrale ─ Audencia, ouverte le 6 septembre 1993, à l'occasion du second prolongement de la ligne ;
 École Centrale ─ Audencia ↔ Bourgeonnière, ouverte le 14 mars 1994, à l'occasion du troisième prolongement de la ligne ;
 Bourgeonnière ↔ Orvault ─ Grand Val, ouverte le 29 août 1994, à l'occasion du quatrième et avant-dernier prolongement de la ligne.

Les terminus réguliers

La ligne 2 compte deux terminus principaux :
 La station Orvault ─ Grand Val, qui constitue le terminus le plus au nord de la ligne, est composée de deux quais encadrant deux voies. Dans le prolongement de la station, se situe un tiroir de retournement à double voie avec aiguillage permettant le retournement des rames.
 La station Gare de Pont Rousseau, qui constitue le terminus le plus au sud de la ligne, est composée de deux quais encadrant trois voies numérotées A, B et C. Avant la station, un aiguillage permet aux rames d'emprunter la bonne voie lors de leur départ. Cette station a été ouverte le 27 août 2007 lors de l'inversion des lignes 2 et 3 au sud-Loire.

Les terminus partiels

La ligne 2 compte deux terminus partiels supplémentaires, qui utilisent les mêmes quais aussi bien pour les rames y faisant terminus que celles ne faisant qu'y passer :
 La station Hôtel Dieu, qui constitue le seul terminus partiel de la ligne situé en plein centre-ville, est composée de deux quais encadrant deux voies. Entre cette station et Aimé Delrue, une voie centrale composée de plusieurs aiguillages permet le retournement des rames. Ce terminus est uniquement utilisé par les renforts d'heures de pointe qui effectuent des départs supplémentaires entre Orvault ─ Grand Val et Hôtel Dieu.
 La station École Centrale ─ Audencia, qui constitue le terminus partiel le plus au nord de la ligne, est composée de deux quais encadrant deux voies. Après la station, en direction d'Orvault ─ Grand Val, un tronçon supplémentaire à voie unique continue tout droit dans la route de la Jonelière, afin que les rames fassent leur demi-tour avant de changer de voie en utilisant l'aiguillage situé entre ce tronçon et la station. Ce terminus partiel est très rarement utilisé, excepté par l'un des premiers services le matin vers Orvault ─ Grand Val.

La ligne 2 emprunte également les infrastructures de la ligne 3 entre Pont Rousseau ─ Martyrs et Neustrie, pour rentrer au dépôt de la Trocardière ou assurer des services supplémentaires sur ce tronçon le matin et le soir :
 La station Espace Diderot, qui constitue le premier terminus partiel sur la ligne 3, est composée de deux quais encadrant deux voies. Au niveau de la station suivante Trocardière, un aiguillage permet aux rames d'accéder au dépôt situé juste à côté. Ce terminus est uniquement utilisé lorsque les rames entrent ou sortent du dépôt, ce qui permet de commercialiser les haut-le-pied de la ligne afin d'éviter que des rames ne circulent à vide.
 La station Neustrie (terminus de la ligne 3), qui constitue le second terminus partiel sur la ligne 3, est composée de deux quais encadrant deux voies. Cette station est desservie par les trois premiers départs et la dernière arrivée de la ligne 2, avant et après les premiers et derniers services de la ligne 3.

Dépôt de la Trocardière

Les rames sont remisées au dépôt de la Trocardière, situé à Rezé à côté de la station Trocardière et de la salle du même nom, et ouvert à l'occasion de la mise en service de la ligne.

Schéma de la ligne

Tracé

Les rames partent du terminus nord Orvault ─ Grand Val puis suivent le boulevard Mendès France avant de bifurquer vers la gauche en direction du pôle d'échanges Le Cardo. La ligne emprunte ensuite le boulevard René-Cassin, puis tourne à droite dans la route de La Chapelle-sur-Erdre après la station René Cassin, puis à gauche dans la rue Eugène Thomas afin de rejoindre la station Chêne des Anglais. Les rames tournent ensuite à gauche dans la rue des Renards, desservent la station Santos Dumont, puis reprennent la rue des Renards en effectuant un léger virage sur la droite. Le tronçon deviens alors un « site banal » et les rames côtoient la circulation générale jusqu'à la station Boissière, avant d'effectuer un virage à gauche en suivant toujours la rue des Renards, puis en rattrapant la rue de la Bourgeonnière à la station Bourgeonnière. Les rames rejoignent ensuite la station Recteur Schmitt, puis suivent la rue Recteur Schmitt puis la route de la Jonelière jusqu'à la station Facultés, avant d'effectuer un virage à gauche dans le boulevard Guy Mollet. Les rames empruntent ensuite le boulevard du Petit-Port, dépassent l'entrée de l'ancien dépôt de la Morrhonnière de l'ancien tramway de Nantes sur la gauche, puis entrent dans le boulevard Michelet. Après la station Saint-Félix, les rames effectuent un léger virage à gauche pour emprunter le boulevard Amiral-Courbet jusqu'à rejoindre l'entrée du pont Général-de-la-Motte-Rouge, avant de tourner à droite sur le quai de Versailles qui longe l'Erdre. Les rames retrouvent ensuite le cours des 50-Otages qu'elles suivent jusqu'à la station Commerce située à côté de l'allée Cassard, en rejoignant la ligne 3 au niveau de la place de l'Écluse.

La ligne continue ensuite tout droit sur le cours Olivier-de-Clisson, en croisant la ligne 1 à la « Croisée des trams » puis en traversant le cours Franklin Roosevelt et l'allée Duguay-Trouin. Elle traverse ensuite le boulevard Jean-Philippot, croise la place Alexis-Ricordeau à gauche, puis longe le boulevard Jean-Monnet et l'Hôtel-Dieu situé à sa droite jusqu'au pont Général-Audibert qu'elle emprunte. La ligne traverse ensuite l'Île de Nantes du nord au sud via le boulevard des Martyrs-Nantais-de-la-Résistance, en croisant la ligne 5 au niveau des boulevards Vincent-Gâche et Babin-Chevaye. Les rames rejoignent ensuite la place Victor-Mangin avant d'emprunter le pont de Pirmil pour rejoindre le pôle d'échanges Pirmil sur la place du même nom. Elles empruntent ensuite le pont des Bataillons FFI pour traverser la Sèvre Nantaise, puis suivent l'avenue de la Libération avant d'effectuer un virage à droite vers la gare de Rezé-Pont-Rousseau, terminus de la ligne.

Liste des stations
Le tableau ci-dessous présente la liste des 25 stations de la ligne du nord au sud. Les terminus sont indiqués en gras.

Exploitation

Principes de la desserte
La ligne fonctionne toute l'année (sauf le 1 mai) du lundi au samedi de  à  et jusqu'à  le vendredi et samedi (un tramway toutes les 10 à 15 minutes le matin, toutes les 3 à 8 minutes en journée, toutes les 10 minutes en soirée et toutes les 30 minutes en service de nuit après ), et le dimanche et jours fériés de  à  (un tramway toutes les 40 à 60 minutes le matin, toutes les 15 à 20 minutes en journée, toutes les 20 minutes en soirée et toutes les 30 minutes en service de nuit après ).

Le matin, les premiers départs s'effectuent à Pont Rousseau ─ Martyrs (en direction de Orvault ─ Grand Val) puisque les haut-le-pied depuis le dépôt de la Trocardière de la ligne empruntent le tronçon de la ligne 3 entre Trentemoult et Pont Rousseau ─ Martyrs. Puis, 3 départs de Neustrie (terminus de la ligne 3) sont effectués par des rames de la ligne en direction de Orvault ─ Grand Val avant les premiers départs de la ligne 3. Les derniers départs s'effectuent aux terminus Gare de Pont Rousseau (du lundi au jeudi) ou Orvault ─ Grand Val (du vendredi au dimanche) et le dernier service de la ligne est prolongé jusqu'à Neustrie afin d'assurer la dernière desserte de cette station avant la fin de service.

En heures de pointe de semaine, des départs supplémentaires sont effectués entre Orvault ─ Grand Val et Hôtel Dieu afin de renforcer la fréquence en centre-ville, mais aussi en direction des campus universitaires nantais.

Pour rentrer ou sortir du dépôt de la Trocardière (situé sur le tracé de la ligne 3), certains services de la ligne 2 empruntent le tronçon sud de la ligne 3 jusqu'à (ou depuis) la station Espace Diderot. Ces services sont ouverts au voyageurs, ce qui permet d'éviter aux rames de circuler à vide aux prises et fin de services.

Temps de parcours moyens
Les temps de trajets indicatifs (arrondis à 5 minutes près) de la ligne 2 depuis la station Orvault ─ Grand Val sont :
 de 15 minutes jusqu'à École Centrale ─ Audencia ;
 de 20 minutes jusqu'à Michelet ─ Sciences ;
 de 30 minutes jusqu'à Commerce et Hôtel Dieu ;
 de 35 minutes jusqu'à Pirmil ;
 de 40 minutes jusqu'à Gare de Pont Rousseau.

La durée d'un trajet entre deux stations est de 2 minutes en moyenne.

Matériel roulant
La ligne 2 est exploitée par la SEMITAN avec des GEC-Alsthom TFS, seules rames que le dépôt de la Trocardière, ou elles sont stockées, peut accueillir.

Tourisme

La ligne 2 dessert, du nord au sud, les lieux d'attraction et monuments suivant :
 la zone et le centre commercial Grand Val ;
 le quartier Nantes Nord ;
 le centre de détention de Nantes ;
 la médiathèque Luce-Courville ;
 le campus du Tetre et ses établissements (Université de Nantes, IAE Nantes, École Centrale de Nantes, Ifremer, Audencia...) ;
 le centre de loisirs du Petit Port et la piscine, la patinoire, le camping, le mini-golf et l'hippodrome ;
 le quai et l'Île de Versailles aux bords de l'Erdre ;
 le pont Saint-Mihiel ;
 le cours et le monument aux 50-Otages ;
 le centre-ville de Nantes ;
 l'Île Feydeau ;
 l'Hôtel-Dieu (un établissement du CHU de Nantes) ;
 l'Île de Nantes ;
 la place Pirmil ;
 l'hôpital Saint-Jacques (un établissement du CHU de Nantes) ;
 la gare de Rezé-Pont-Rousseau.

Projets

Une interconnexion est en projet avec la ligne 1 au nord de l'agglomération, à partir de la station Recteur Schmitt. La desserte entre Morrhonnière ─ Petit Port et Haluchère ─ Batignolles se fera par l'intermédiaire de ce nouveau tronçon sera donc communs aux deux lignes, dont le premier segment sur la ligne 1 a été mise en service le  entre Haluchère ─ Batignolles et Ranzay. Avec le lancement des 3 nouvelles lignes de tramway, c'est finalement la future ligne 6 qui empruntera cette connexion (et non pas les lignes 1 et 2 comme prévu initialement).

Notes

Références

Voir aussi

Articles connexes
 Semitan
 Tramway de Nantes
 Liste des stations du tramway et du Busway de Nantes
 Histoire des transports en commun à Nantes

Liens externes
 Site de la TAN

Nantes, 2
Tramway de Nantes

redpajama_set_name: RedPajamaWikipedia


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