\section{Results}\label{sec:results}

In the following we want to compare the performance between the conventional system and the \ac{AE}. We therefore start with a sanity check using two simplified channel models, one consisting only of \ac{AWGN}, denoted as $\mathcal{C}_\mathrm{A}$, and another with \ac{AWGN} and \ac{CD}, denoted as $\mathcal{C}_\mathrm{AD}$. In the beginning we set $N_\mathrm{adj}=0$ to ensure that all compensation has to be performed by the transmitter. As we know perfect transmission systems (achieving capacity limits) for those two channel-models, we can check whether the AE is able to also approximate those perfect solutions. The resulting SE is shown in Fig. \ref{fig:ae-performance}. As the \ac{AE} achieves the performance of the conventional system for $\mathcal{C}_\mathrm{A}$ and $\mathcal{C}_\mathrm{AD}$ this means that, \emph{jointly},
\begin{enumerate}
	\item a constellation that maximizes average distance between \ac{IQ}-symbols, and,
	\item a Nyquist-pulse-shaping, as well as
	\item a perfect \ac{CD}-compensation
\end{enumerate}
have been learned! Further, there is, of course, no gain compared to the conventional implementation. Nevertheless, this leads to the hypothesis that the AE is (at least in some cases) able to learn perfect solutions.

Applying the \ac{AE} to the channel-model $\mathcal{C}_\mathrm{ADN}$ (consisting of noise, \ac{CD} and \ac{KNL} as described in Section \ref{sec:system}) and starting with $N_\mathrm{adj}=0$ results in a small performance gain in terms of $SE$ compared to the conventional system only (see Fig. \ref{fig:ae-performance}). As we have chosen a classical communications structure we can now analyze the single blocks of the AE and hence try to find the source of this gain. 


We first analyze the mapping of the \ac{AE} of Fig. \ref{fig:constellation}, which is indeed a more efficient constellation compared to a conventional 256-\ac{QAM} in case the $SE$ is calculated on uncoded symbols. Its circular shape can maximize the average distance between neighboring symbols. Furthermore, the density of symbols is lower in the high power than in the low power regime, tolerating higher \ac{KNL}-induced phase-shifts. This also matches with the results of \cite{jones2019end}. As both facts may already lead to a slightly higher performance we, nevertheless, analyze the surprising pulse shaping filter $\mathbf{F}$ of Fig. \ref{fig:ae-pulse-shapes-zero}, where a dark blue depicts the squared amplitude (power) of its impulse response, red depicts the received output pulse (after the channel) and dotted black a reference Nyquist-pulse. The same holds for its corresponding phase in blue, orange and gray.


In Fig. \ref{subfig:zero-amplitude-low} the signals' squared amplitudes for low powers ($P=-10\,\mathrm{dBm}$) are shown. Here, one can see that the \emph{received} output signal $\mathbf{y}$ matches with the well-known Nyquist-pulse perfectly. Hence, the AE shaped the input signal $\mathbf{x}$ such that it obtains orthogonal samples in time at the output, to not lose information by the implemented sampling (as $N_\mathrm{adj}=0$). Fig. \ref{subfig:zero-phase-low} further depicts the phase response of $\mathbf{F}$ that matches with the inverse of the well-known phase of \ac{CD} (wrapped parabolic) and, hence, a classic \ac{CD}-compensation, almost perfectly. The same holds for $P=0\,\mathrm{dBm}$ in Fig. \ref{subfig:zero-amplitude-mid} and \ref{subfig:zero-phase-mid}. At higher powers, where the AE-performance in terms of \ac{SE} follows the same curve as for the conventional system and especially at $P=10\,\mathrm{dBm}$, we experience a missing convergence of the \ac{AE} resulting in Fig. \ref{subfig:zero-amplitude-high} and \ref{subfig:zero-phase-high}. There, no sensible pulse-shapes were learned, that one can interpret. This may have two reasons, while we are not able to exclude the one or the other: First, the gradient may get lost such that optimization fails or, second, there is no proper constellation-pulse-shape-combination that works at high powers. Only by increasing $N_\mathrm{adj}$ we can see in Fig. \ref{fig:ae-performance} a significant performance gain from $1.32\,\mathrm{\sfrac{bits}{s \cdot Hz}}$ at $P=-10\,\mathrm{dBm}$ to $4.46\,\mathrm{\sfrac{bits}{s \cdot Hz}}$ at $P=0\,\mathrm{dBm}$, which increases with $N_\mathrm{adj}$ until it saturates at ca. $N_\mathrm{adj,opt}=20$. For higher values of $N_\mathrm{adj}$ there is no further gain (see Fig. \ref{fig:ae-performance} and $N_\mathrm{adj}=40$). Again, we conduct an analysis of the single blocks of the AE to search for the source of this gain. It turned out, that the constellations, generated by the mapping $\mathbf{W}$, again converge to the same constellation, shown in Fig. \ref{fig:constellation}. For low powers, there is no perfect convergence\footnote{As there is no gain for the AE being more precise or smoother than the channel's induced noise.} and hence, the constellation-diagram is not that symmetric, while still showing the same circular structure. We also interpret the learned pulse-shapes for $N_\mathrm{adj,opt}=20$ starting with their \ac{PSD} as shown in Fig. \ref{fig:psd-ae}. 


Here, one can see for the input power with the highest gain (compared to classic systems) at $P=0\,\mathrm{dBm}$, that the \ac{AE} achieved a pulse-shaping that has much less spectral broadening by achieving a high \ac{DC}-offset (or a remaining \ac{CD}, respectively) at the output signal $\mathbf{y}_\mathrm{o}$. Unfortunately, it was not able to achieve the same for an even higher power $P=10\,\mathrm{dBm}$. Nevertheless, this leads to the hypothesis that the \ac{AE} tries to level the derivative of the signal's squared amplitude (power) and apply a phase modulation, to avoid \ac{KNL} in general. The corresponding filter shapes are shown in Fig. \ref{fig:ae-pulse-shapes-twenty}. All amplitudes at the three different power levels in Figs. \ref{subfig:twenty-amplitude-low}, \ref{subfig:twenty-amplitude-mid} and \ref{subfig:twenty-amplitude-high} do not provide any deeper insights. In contrast, the phase responses do: For low powers, one can still identify some wrapped parabolic shape of the learned pulses (see Fig. \ref{subfig:twenty-phase-low} and \ref{subfig:twenty-phase-mid}). Surprisingly, this is not only the case for the learned but also for the received phase. This means that the \ac{AE} left some \ac{CD} uncompensated before the pulse is transmitted over the channel. Unfortunately, this does not hold for high powers in Fig. \ref{subfig:twenty-phase-high}.
{"config": "arxiv", "file": "2006.15027/content/results.tex"}