\chapter{Spectrally efficient communications over the satellite channel}\label{ch:spec_tpack}


\lettrine{I}{n} this chapter, we apply the time-frequency packing (TF packing) technique to nonlinear satellite channels. In particular, we design highly efficient schemes by choosing 
the time and frequency spacings which give the maximum value of SE. 
We assume a realistic satellite channel where nonlinear distortions originate from the presence of a high-power amplifier (HPA).
The considered system is also affected by intersymbol interference (ISI), due to the presence of input and output multiplexing (IMUX and OMUX) filters placed before and after the HPA and intentionally introduced by the adoption of the time packing technique 
as well. We limit our investigation to systems in single-carrier-per-transponder operation (i.e., each transponder is devoted to the amplification of the signal coming from only one user\footnote{However, the actual 
transponders can be hosted on two or more co-located satellites.}). Although interchannel interference (ICI) is also present due to frequency packing of signals coming from different transponders, a single-user detector is considered at the receiver. 
We consider two different approaches 
to detection for nonlinear channels,
namely the use of a detector taking into account the nonlinear effects and a more 
traditional scheme based on predistortion and memoryless detection.
In the case of predistortion, we consider the dynamic data predistortion technique described in~\cite{KaSa91,CaDeGi04}, whereas in the case of advanced detection we 
consider the receiver described in the previous chapter employing the channel shortening technique
(CS) for nonlinear satellite channels. It should be noted that we apply the TF
packing technique to nonlinear satellite channels for which, usually,  even by using RRC pulses there is still ISI at the receiver.

As mentioned, with respect to Chapter~\ref{chap:time_pack}, we here consider nonlinear satellite channels instead of linear ones. 


Our aim here is to show the benefits that can be obtained by employing time-frequency packing jointly with a channel shortening 
receiver.


The proposed TF packing technique promises to provide increased SEs at least for low-order modulation formats. In fact, when dense constellations with shaping are employed, 
we fall in a scenario similar to that of the Gaussian channel with Gaussian inputs for which orthogonal signaling with no excess bandwidth (rectangular shaping pulses) is optimal (although this is mainly true for the linear channel 
and not in the presence of a nonlinear HPA, 
since shaping increases the peak-to-average power ratio). Improving the achievable SE without increasing the constellation order can be considerably convenient since it is well known that low-order constellations are more robust 
to channel impairments such as time-varying phase noise and non-linearities. It is expected that the use of low-order modulations in conjunction with TF packing provides similar advantages in terms of robustness against channel 
impairments.

The proposed approach to improve the SE is very general and the  case of satellite systems for broadband and broadcasting applications must be thus considered just as an example to illustrate the benefits that can be obtained 
through the application of the TF packing paradigm coupled with advanced receiver processing.

The remainder of this chapter is organized as follows. In \S\ref{s:system_model}, we introduce the system model. 
The framework that we use to evaluate the SE of satellite systems is detailed in \S\ref{s:spectral efficiency}, whereas different approaches to the detection 
for the considered channel are described in \S\ref{s:auxiliary channels}. Numerical results are reported in \S\ref{s:results}, where we show how the 
proposed technique can improve the SE of DVB-S2 systems. Finally, conclusions are drawn in \S\ref{s:conclusions}.


\section{System Model}~\label{s:system_model}
We consider the forward link of a transparent satellite system, where synchronous users employ the same linear modulation format, shaping pulse $p(t)$, and symbol interval 
(or time spacing) $T$, and access the channel according to a frequency division multiplexing scheme. 
The transmitted signal in the uplink can be expressed as 
\begin{equation}\label{e:transmitted signal}
x(t)=\sum_\ell \sum_k c^{(\ell)}_{k} p(t-kT)e^{j2\pi \ell F_u t}\, ,
\end{equation}
where $c^{(\ell)}_{k}$ is the symbol transmitted by user $\ell$ during the $k$-th symbol interval, and $F_u$ is the frequency spacing between adjacent channels.\footnote{In this scenario, we will use the terms,
``channels'', ``users'', and ``carriers'' interchangeably.}
The transmitted symbols belong to a given zero-mean $M$-ary complex constellation. Notice that, in order to leave out border effects, the summations 
in~\eqref{e:transmitted signal} extend from $-\infty$
to $+\infty$, namely an infinite number of time epochs and carriers
are considered. 
In \mbox{DVB-S2} standard the base pulse $p(t)$ is an RRC-shaped pulse with roll-off factor $\alpha$ (equal to 0.2, 0.3, or 0.35 depending on the service requirements). We denote by $W$ the bandwidth of pulse $p(t)$. In case 
of pulses employed in DVB-S2, it is $W=(1+\alpha)/T$ since orthogonal signaling is considered. When time packing is adopted, $W$ becomes a further degree of freedom, as described later.


As commonly assumed for broadband and broadcasting systems, on the feeder uplink (between the gateway and the satellite) the impact of thermal noise can be neglected due to a high transmit signal strength. Hence, in our analysis, 
we have considered a noiseless feeder uplink.
Although the TF packing can be applied to other and more general scenarios,  we consider here a single-carrier-per-transponder scenario, where different carriers undergo independent amplification by different transponders on board 
of satellite, each of which works with 
a single carrier occupying its entire bandwidth. This case is
particularly relevant for digital broadcasting services since it allows a more efficient use of on-board resources (in particular the HPA can work closer to saturation).\footnote{The multiple-carriers-per-transponder 
scenario is conceptually similar to that considered in this chapter, the only difference being the fact that more adjacent carriers are amplified by the same HPA. Thus, the effects of nonlinear ICI (intermodulation distortion) 
become more relevant and proper multicarrier detection or predistortion algorithms could be also considered~\cite{BeSe10,Be11,PiShZeRoGrZiGrHeCi12}.}  
The transponder model for user $\ell$ is composed of an IMUX filter which selects the $\ell$-th carrier, an HPA, 
and an OMUX filter which reduces the out-of-band power due to the spectral regrowth after nonlinear amplification~\cite{DVB-S2-TR}.
The HPA is a nonlinear memoryless device defined through its AM/AM and AM/PM characteristics, describing the amplitude and phase distortions caused on the signal at 
its input. 

\begin{figure}
	\begin{center}
		\includegraphics[width=88mm]{images/sat_channel/system_model2} 
		\caption{System model.{\label{f:system_model}}} 
	\end{center}
\end{figure}

The outputs of different transponders are multiplexed again in the downlink to form the signal $s(t)$, and we assume that the adjacent users have a frequency 
separation of $F_d$, usually equal to that in the uplink. Fig.~\ref{f:system_model} shows the system model highlighting the satellite transponder for user with $\ell=0$.
The useful signal at the user terminal is still the sum of independent contributions, one for each transponder (although these contributions are no more, rigorously, 
linearly modulated due to the nonlinear transformation of the on-board HPA).
The received signal is also corrupted by the downlink AWGN, whose low-pass equivalent  $w(t)$ 
has power spectral density (PSD) $N_{0}$. The low-pass equivalent of the received signal has thus expression
\begin{equation}
 r(t)=s(t)+w(t) \, .
\end{equation}
We remark that, in the simulation results, this system has been simulated with realistic assumptions. In other words, the satellite receives the entire signal (\ref{e:transmitted signal}). 
Each carrier is then selected by the IMUX filter of its own transponder,\footnote{Being the IMUX a non ideal filter and due to a possible overlap among different carriers, 
this filtering will not be perfect and ICI will occur.} amplified by its own HPA, filtered again by the OMUX filter, and then the signals at the output of all transponders are multiplexed again on air.

We evaluate the ultimate performance limits of this communication system when single-user detection is employed at the receiver side. 
The proposed technique consists of allowing interference in time and/or frequency by reducing the values of $T$, $F_u$, and $F_d$, (partially) coping with it at the 
receiver, in order to increase the SE. 
In other words, $T$, $F_u$, and $F_d$ are chosen as the values that give the maximum value of the SE. 
These values depend on the employed detector---the larger the interference that the receiver can cope with, the larger the SE and the lower the values of time and 
frequency spacings.
Notice that, since we are considering single-user detection, the receiver is not able to deal with the interference due to the overlap of different channels.
In this case, the optimization of the frequency spacings is actually an optimization of the frequency guard bands generally introduced in satellite systems to avoid 
the nonlinear cross-talk, since a single-user receiver can tolerate only a very small amount of ICI.

The considered nonlinear satellite channel reduces, as a particular case, to the linear channel, provided that the HPA is driven far from saturation. Hence, all the considerations in this chapter can be straightforwardly 
extended to the linear channel case. A few results can be found on Chapter~\ref{chap:time_pack}.


\section{Optimization of the spectral efficiency}\label{s:spectral efficiency}
We describe the framework used to evaluate the ultimate performance limits of the considered satellite system and to perform the optimization of the time and 
frequency spacings. 
To simplify the analysis, we will assume $F_u=F_d=F$. 
We perform this investigation by constraining the complexity of the employed receiver. In particular, as mentioned, we assume that a single-user detector is used. 
For this reason, without loss of generality we only consider the detection of symbols $\bm{c}^{(0)}=\{c^{(0)}_k\}$ of user with $\ell=0$.\footnote{Assuming a system with 
an infinite number of users, the results do not depend on a specific user.} In addition, we also consider low-complexity receivers taking into account only a portion 
of the actual channel memory. Under these constraints, we compute the IR, i.e., the average mutual information when the channel inputs are 
i.u.d. random variables belonging to a given constellation. Provided that a proper \textit{auxiliary channel} can be defined 
for which the adopted low-complexity receiver is optimal, the computed IR represents an achievable lower bound of the IR of the actual channel, according to 
mismatched detection~\cite{MeKaLaSh94}.


Denoting by $\bm{r}^{(0)}$  a set of sufficient statistics for the detection of $\bm{c}^{(0)}$, the achievable IR, measured in bit per channel use, can be obtained as
\begin{equation}\label{e:IR}
I_{\mathrm{R}} =\lim_{N\rightarrow \infty}\frac{1}{N} E
\Bigg\{\log \frac{q(\bm{r}^{(0)} \arrowvert \bm{c}^{(0)} )}{q(\bm{r}^{(0)})}  \Bigg\}\, ,
\end{equation}
where $N$ is the number of transmitted symbols. 
The probability density functions $q(\bm{r}^{(0)} \arrowvert \bm{c}^{(0)})$
and $q(\bm{r}^{(0)})$ are computed by using the optimal maximum-a-posteriori (MAP) symbol detector for the auxiliary channel, while the expectation in~(\ref{e:IR}) 
is with respect to the input and output sequences generated according to the actual channel model~\cite{ArLoVoKaZe06}. In the next section, we will discuss two 
different low-complexity detectors for nonlinear satellite channels and we will define the corresponding auxiliary channels.

We can define the user's bandwidth as the frequency separation $F$ between two adjacent carriers. The achievable SE is thus
\begin{equation}
\eta=\frac{I_{\mathrm{R}}}{FT}  \quad[
\textrm{b/s}/\textrm{Hz} ] . \label{e:eta}
\end{equation}
The aim of the proposed technique is to find the values of $F$ and $T$ providing, for each value of the signal-to-noise ratio (SNR), the maximum value of SE achievable by that particular receiver, optimal for the considered specific 
auxiliary channel. Namely, we compute
\begin{equation}
\eta_\text{M}=\max_{F,T>0} \eta(F,T) \, . \label{e:eta_M}
\end{equation}
Typically, the dependency on the SNR value is not critical, in the sense that we can identify, for each shaping pulse and modulation format, two or at most three SNR regions for which the  optimal spacings practically have the same 
value. 

Having removed the constraint of orthogonal signaling,
one more degree of freedom in the SE optimization is represented by the bandwidth $W$ of the shaping pulse $p(t)$. Hence, guided by the same idea behind the TF packing technique, we can also optimize $W$, further increasing both ICI 
and ISI due to the adjacent users and to the IMUX and OMUX filters, respectively.
Whereas on the AWGN channel this optimization is implicit in TF packing, in the sense that we can obtain the same ICI by fixing $F$ and increasing $W$ or by fixing $W$ and decreasing $F$, 
this is no more true for our nonlinear channel since IMUX and OMUX bandwidths are kept fixed. Hence, an increased value of $W$ also increases the ISI.
The benefit of the bandwidth optimization is twofold: it can be used  as an alternative to frequency packing (e.g., in cases where the transponder frequency plan cannot be modified and hence frequency packing is not an option),
or it can be used to improve the results of TF packing. In this case, we thus compute 
\begin{equation}
\eta_\text{M}=\max_{F,T,W>0} \eta(F,T,W) \, . \label{e:eta_M2}
\end{equation} 

For fair comparisons in terms of SE, we need a proper definition of the SNR. We define the SNR as the ratio  $P_{\mathrm{sat}}/ N_0F$ between the peak power $P_{\mathrm{sat}}$ at the output of an  
HPA in response to a continuous wave input, denoted as the amplifier saturation power, and the noise power in the bandwidth assigned to each carrier, which coincides with the frequency spacing $F$ between two adjacent carriers. 
This is because  in a satellite forward link, the two main resource constraints are the available frequency spectrum and the radiated power on-board of the satellite.
The adopted SNR definition is independent of the transmit waveform and its parameters. This provides a common measure to compare the performance of different solutions in a fair manner. 
In the following, we also define the output back-off (OBO) for each waveform (or modulation scheme) as the power ratio (in dBs) between the unmodulated carrier at saturation and the modulated carrier after the OMUX. 

Without loss of generality, $T$ and $F$ in (\ref{e:eta})-(\ref{e:eta_M2}) can be normalized to some reference values $T_B$ and $F_B$.
We will denote $\nu=F/F_B$ and $\tau=T/T_B$. In the numerical results, we will choose $T_B$ and $F_B$ as the symbol time and the frequency spacing adopted in the DVB-S2 standard, which is considered as a benchmark scenario. 

\section{Considered detectors and corresponding auxiliary channel models}\label{s:auxiliary channels}

The system model described in~\S\ref{s:system_model} is representative of the considered scenario and has been employed in the information-theoretic analysis and in the simulations results. In this section, we describe 
the employed auxiliary channel models and the corresponding optimal MAP symbol detectors. As explained in~\S\ref{s:spectral efficiency}, they are used to compute two lower bounds on the SE for the considered 
channel~\cite{ArLoVoKaZe06}. Since these lower bounds are achievable by those receivers, we will say that the computed lower bounds are the SE values of the considered channel when those receivers are employed.

\subsection*{Memoryless model and predistortion at the transmitter}\label{ss:predistortion}
When the HPA AM/AM and AM/PM characteristics are properly estimated and fed back to the transmitter, the sequence of symbols $\{c^{(\ell)}_k\}$ can be properly 
predistorted to form the sequence  $\{c'^{(\ell)}_k\}$ that is transmitted instead, in order to compensate for the effect of the non-linearity and possibly to reduce the ISI. 
Here we consider the dynamic data predistortion technique described in~\cite{KaSa91,CaDeGi04} and also suggested for the application in DVB-S2 systems~\cite{DVB-S2-TR}, where the symbol $c'^{(\ell)}_k$ transmitted by user $\ell$ 
at time $k$ is a function of a sequence of $2L_p+1$ input symbols, i.e.,  $c'_k=f(c_{k-L_p},\ldots,c_k,\ldots,c_{k+L_p})$. 
The mapping  $f$ at the transmitter is implemented through a look-up table (LUT), which is computed through an iterative
procedure performed off-line and described in~\cite{KaSa91,CaDeGi04} for each modulation format, setting of the system parameters, and SNR value.  This procedure searches the best trade-off
between the interference reduction and the increase of the OBO.
The complexity at the transmitter depends on the number of symbols accounted for through the parameter $L_p$. 
The transmitted signal is thus
\begin{equation}
 x(t)=\sum_\ell\sum_k c'^{(\ell)}_kp(t-kT)e^{j2\pi\ell Ft}
\end{equation}
whereas, at the receiver, a simple single-user memoryless channel is assumed corresponding to the auxiliary channel (for user with $\ell=0$)
\begin{equation}\label{e:auxiliary_channel_pred}
  r^{(0)}_k=c^{(0)}_k+n_k
\end{equation}
where $n_k$ is a zero-mean circularly symmetric white Gaussian
process with PSD $(N_0+N_I)$, $N_I$ being a design parameter which
can be optimized through computer simulations---an increase of the assumed noise variance can improve 
the computed achievable lower bound on the SE~\cite{BaFeCo09b}.


\subsection*{Model with memory and advanced detection}\label{ss:advanced_detection}
A valid alternative to nonlinear compensation techniques at the transmitter relies upon the adoption of advanced detectors which can manage the nonlinear 
distortions and the ISI. 
In this work, we consider detection based on an approximate signal model described in~\cite{CoPi12}, which comes from a simplified 
Volterra series expansion of the nonlinear channel. 
To limit the receiver complexity with a limited performance degradation, we also apply a CS technique~\cite{FaMa73}.
In fact, when the memory of the channel is too large to be taken into account by a full complexity detector, an excellent performance can be achieved 
by properly filtering the received signal before adopting a reduced-state detector~\cite{FaMa73}.

A very effective CS technique for general linear channels is described in~\cite{RuPr12}, while its extension to nonlinear satellite systems is reported 
in Chapter~\ref{ch:cs_sat}. 
We will denote the memory taken into account by the advanced detection scheme as $L_r$.

\section{Numerical results}\label{s:results}
\subsection*{Spectral efficiency}
Considering the DVB-S2 system as a benchmark scenario, we now
show the improvement, in terms of SE, that can be obtained by adopting the TF packing technique joint with an advanced processing at the receiver. In the following, we assume as reference the time and frequency spacings 
of DVB-S2, i.e., $1/T_B=27.5$~Mbaud and $F_B=41.5$ MHz, and use these values to normalize all time and frequency spacings. Let us consider a typical DVB-S2 scenario where, at the transmitter, the shaping 
pulse $p(t)$ has a RRC spectrum, whereas the IMUX and OMUX filters and the nonlinear characteristics of the HPA are those reported in~\cite[Figs.~H.12 and H.13]{DVB-S2-TR}. 
The standard considers the following modulation formats: QPSK, 8PSK, 16APSK, and 32APSK. 
To combat ISI and nonlinear distortions, a data predistorter is employed at the transmitter whereas at 
the receiver a symbol-by-symbol detector is assumed.
Here, we consider the predistorter described in~\S\ref{ss:predistortion}, with $L_p=2$ for QPSK, 8PSK 16APSK, and $L_p=1$ for 32APSK.
The SE results have been obtained by computing the IR in~(\ref{e:IR}) by means of the Monte Carlo method described in~\cite{ArLoVoKaZe06}. For each case, i.e., for each modulation format, employed detection algorithm, 
and choice of the system parameter, we also performed a coarse optimization of the noise variance to be set at 
receiver~\cite{BaFeCo09b}, 
and of the amplifier operation point through the OBO. Unless otherwise specified, the roll-off factor of the RRC pulses is $\alpha=0.2$, which is the lowest value considered in the standard. 

We first consider the achievable SE of our benchmark scenario, and in Fig.~\ref{f:ASE_DVB} we report $\eta$ as a function of $P_{\mathrm{sat}} /N_0 F$ for the four modulation formats of the standard (QPSK, 8PSK, 16APSK, and 32APSK). 
In this case, it is $W=(1+\alpha)/T_B$.
We verified that comparable SE values can be also obtained by using, instead of the predistorter, the advanced detection scheme of~\S\ref{ss:advanced_detection} with $L_r=0$ (MMSE detection). We also consider two alternative 
ways that, at least in the case of a linear channel, can be used to improve the SE without resorting to TF packing.\footnote{On a nonlinear satellite channel, due to the increased peak-to-average power ratio, their application must 
be carefully considered since not necessarily produces the expected benefits.} The simplest approach relies on the increase of the modulation cardinality, and in Fig.~\ref{f:ASE_DVB} we also show the SE for the 64APSK 
modulation~\cite{LiAl08}.
It can be seen that the 64APSK modulation, due to the higher impact of the non-linearities, allows to increase the SE only at high SNR values and it seems there is no hope to improve the SE in the low and medium SNR regions.

\begin{figure}
	\begin{center}
		\includegraphics[width=88mm]{images/sat_channel/ase_orth_v2}
		\caption{Spectral efficiency of DVB-S2 modulations with roll-off 0.2, data predistortion, and memoryless detection. Comparison with a constellation of increased cardinality (64APSK).}\label{f:ASE_DVB} 
	\end{center}
\end{figure}

An alternative way of improving the SE is based on a reduction of the roll-off factor. In Fig.~\ref{f:ASE_TF_pre}, we consider QPSK and 16APSK modulations in a scenario where predistortion at the transmitter and symbol-by-symbol 
detection at the receiver are still employed. We show the SE improvement that can be obtained by reducing the roll-off to $\alpha=0.05$.\footnote{We properly modify the transmitted signal such that it occupies the same bandwidth 
as that of the signal with roll-off 0.2. We verified that, in this particular case, no improvement can be obtained by resorting to a more sophisticated receiver based on linear or nonlinear equalization in addition to or in 
substitution of the predistorter.}
We can observe that the roll-off reduction improves the SE with respect to DVB-S2 for all SNR values. On the other hand, as shown in Fig.~\ref{f:ASE_TF_pre}, better results can be obtained by allowing 
TF packing. The values of $T$ and $F$ are chosen as those providing the largest SE. This search is carried out by evaluating~(\ref{e:eta_M}) on grid of values of $T$ and $F$ (coarse search), followed by interpolation 
of the obtained values (fine search). We point out that these curves have been obtained without 
reducing the roll-off factor, which is still $\alpha=0.2$, and employing the same predistorter and symbol-by-symbol receiver adopted in the DVB-S2 system~\cite{KaSa91,CaDeGi04}. 

\begin{figure}
	\begin{center}
		\includegraphics[width=88mm]{images/sat_channel/ASE_nonlin_TF_v2}
		\caption{Improvements, in terms of spectral efficiency, that can be obtained for QPSK and 16APSK modulations by reducing of the roll-off ($\alpha=0.05$) or by adopting the TF packing technique. 
		In all cases, predistortion at the transmitter and memoryless detection at the receiver are employed.}\label{f:ASE_TF_pre} 
	\end{center}
\end{figure}

With the aim of further improving the performance, we now consider TF packing and a system without predistortion at the transmitter but using the advanced detection algorithm described in~\S\ref{ss:advanced_detection}, joint 
with CS  ($L_r=1$, to reduce the receiver complexity, since in this case the BCJR algorithm has only $M$ states). 
The assumed order of the Volterra model~(\ref{eq:volterra_apprx}) is $v=5$ since, in this case, when considering a single-carrier transmission (no adjacent users) the minimum mean square error between the model and the actual 
signal is very low. The results for QPSK and 16APSK modulations are reported in Fig.~\ref{f:ASE_TF_CS}, where we also show the DVB-S2 benchmark curves  discussed above and the curves related to TF packing 
when predistortion at the transmitter and memoryless detection at the receiver are used. These results show the impressive improvement achievable by TF packing combined with the considered advanced receiver, which, with a memory of 
only one symbol, can cope with much more interference than the schemes employing the predistorter and a memoryless detector.


\begin{figure}
	\begin{center}
		\includegraphics[width=88mm]{images/sat_channel/ASE_nonlin_L1_v2}
		\caption{Spectral efficiency for QPSK and 16APSK modulations with TF packing and advanced detector (TF pack., adv. det.). Comparison with the DVB-S2 scenario and the case of TF packing when a predistorter and a symbol-by-symbol detector are adopted (TF pack., pred.).}\label{f:ASE_TF_CS} 
	\end{center}
\end{figure}

In the previous figures we considered, as mentioned, a bandwidth $W=1.2/T_B$. We also considered the possibility of optimizing the bandwidth $W$, as described in~\S\ref{s:spectral efficiency}.
In Fig.~\ref{f:ASE_B_PACK}, we consider QPSK modulation and the advanced receiver with $L_r=1$. As expected, the combination of TF packing with the bandwidth optimization gives the best results. We also show the results in case only 
time packing or only the bandwidth optimization are adopted. Interestingly, the SE of time packing with bandwidth optimization is quite similar to that achievable by TF packing.

\begin{figure}
	\begin{center}
		\includegraphics[width=88mm]{images/sat_channel/ase_b_pack_v2}
		\caption{Spectral efficiency of QPSK modulation with TF packing and bandwidth optimization by adopting the advanced receiver with CS ($L_r=1$).}\label{f:ASE_B_PACK}
	\end{center}
\end{figure}

Finally, to summarize the results, Figure~\ref{f:ASE_TFB} shows the SE for all DVB-S2 modulations (QPSK, 8PSK, 16APSK and 32APSK) with TF packing, bandwidth optimization, and the advanced receiver. For clarity, 
we show only one curve which, for each abscissa, reports only the largest value of the four curves (the ``envelope''). In the same figure, we also plot three other SE curves obtained by using predistortion and a memoryless receiver. 
The lowest one is that corresponding to the DVB-S2 scenario (one curve which is the ``envelope'' of all four curves in Fig.~\ref{f:ASE_DVB}), the SE curve for the 64APSK modulation, and the SE curve in  case of roll-off $\alpha=0.05$ 
reduction. In this latter case, we considered all modulations with cardinality up to 64, and hence this curve represents the effect 
of both roll-off reduction and cardinality increase with respect to DVB-S2. The figure shows that TF packing and advanced receiver processing allows a SE improvement of around 40$\%$ w.r.t. DVB-S2 at high SNR.
This gain is partly due to the fact that current DVB-S2 standard does not support higher order modulations or lower roll-off values. However, there is still considerable SE improvements at lower SNR values.


\begin{figure}
	\begin{center}
		\includegraphics[width=88mm]{images/sat_channel/ase_TFB_v2}
		\caption{Spectral efficiency of TF packing with bandwidth optimization (TF pack., $W$ opt.). Comparison with DVB-S2, 64APSK and roll-off reduction.}\label{f:ASE_TFB}
	\end{center}
\end{figure}

\subsection*{Modulation and coding formats}
What information theory promises can be approached by using proper coding schemes. All the considered modulation and coding formats (MODCODs) use the low-density parity-check (LDPC) codes with length 64800 bits of the DVB-S2 
standard.  We adopt the optimized values for $T$, $F$, and $W$ and the advanced detector described in~\S\ref{ss:advanced_detection}.
Due to the soft-input soft-output nature of the considered detection algorithm, we can adopt iterative detection and decoding. We distinguish between local iterations, within the LDPC decoder, and global iterations,
between the detector and the decoder. Here, we allow a maximum of 5 global iterations and 20 local iterations.

BER results have been computed by means of Monte Carlo simulations and are reported in the
SE plane in Figure ~\ref{f:modcods} using, as reference, a BER of $10^{-6}$. In the same figure, the performance of the DVB-S2 MODCODs is also shown for comparison. 
We recall that for them predistortion at the transmitter and symbol-by-symbol detection at the receiver are adopted. Moreover, for them we have $\tau=1$ and $\nu=1$.
The details of the considered MODCODs are reported in Tables~\ref{t:modcods_det} and~\ref{t:modcods_dvbs2}.
These results are in perfect agreement with the theoretical analysis and confirm that the TF packing technique can provide an impressive performance improvement w.r.t. the DVB-S2 standard. 

\begin{figure}
	\begin{center}
		\includegraphics[width=88mm]{images/sat_channel/modcods_v2}
		\caption{Modulation and coding formats of the DVB-S2 standard and comparison with those designed for the proposed TF packing technique with optimized bandwidth.}\label{f:modcods}
	\end{center}
\end{figure}

\begin{table}
\caption{Details of the MODCODs based on TF packing.}\label{t:modcods_det}
    \centering
    \begin{tabular}{c|c|c|c|c|c|c|}
	\cline{2-7}
	& rate & $\tau$ & $\nu$ & $W_{\rm opt}$  &  $\!P/N_0 F\!$  & $\eta$ \\
	&   &   &   &    &    [dB] &   [b/s/Hz]\\
	\hline
	\hline
	\multirow{3}{*}{QPSK} & 1/3 & 0.833  & 1.00 & +20\% & -1.7 & 0.53 \\
	& 1/2 & 0.750 & 0.90 & +20\% & 2.2 & 0.98  \\
	& 3/5 & 0.750 & 0.90 & +20\% & 3.6 & 1.18 \\ 
	\hline
	\multirow{3}{*}{8PSK} & 1/2 & 0.731 & 0.95 & +30\%  & 5.3 & 1.43  \\
	& 3/5 & 0.731 & 0.95 & +30\%  & 7.4 & 1.72  \\
	& 2/3 & 0.731 & 0.95 & +30\% & 8.5 & 1.91  \\
	\hline
	\multirow{2}{*}{16APSK} & 2/3 & 0.792 & 0.90 & +20\% & 11.1 & 2.48  \\
	& 3/4 & 0.750 & 0.90& +20\%  & 14.1 & 2.94  \\
	\hline
	\multirow{4}{*}{32APSK} & 2/3 & 0.731 & 0.95 & +30\% & 15.3 & 3.18  \\
	& 3/4 & 0.731 & 0.95 & +30\%  & 17.5 & 3.58 \\ 
	& 5/6 & 0.731 & 0.95 & +30\%  & 19.5 & 3.98 \\ 
	& 8/9 & 0.731 & 0.95 & +30\%  & 21.2 & 4.24 \\ 
	\hline
    \end{tabular}
\end{table}


\begin{table}
   
     \caption{Details of the DVB-S2 MODCODs. In this case, $\tau=1$ and $\nu=1$.}\label{t:modcods_dvbs2}
     \centering
    \begin{tabular}{c|c|c|c|}
	\cline{2-4}
	& rate &  $P/N_0 F$ [dB] & $\eta$ [b/s/Hz]\\
	\hline
	\hline
	\multirow{3}{*}{QPSK} & 1/2 & 0.1 & 0.66 \\
	& 3/5  & 1.4 & 0.79  \\
	& 3/4 & 3.2 & 0.99 \\ 
	\hline
	\multirow{3}{*}{8PSK} & 3/5 & 4.6 & 1.19 \\
	& 3/4 & 7.3 & 1.49  \\
	& 8/9 & 10.0 & 1.77  \\
	\hline
	\multirow{2}{*}{16APSK} & 3/4 & 10.9 & 1.99  \\
	& 4/5 & 12.0 & 2.12  \\
	& 5/6 & 12.7 & 2.21  \\
	& 8/9 & 14.6 & 2.35  \\
	\hline
	\multirow{4}{*}{32APSK} & 3/4  & 14.3 & 2.48  \\
	& 4/5 & 15.6 & 2.65 \\ 
	& 5/6 & 16.5 & 2.76 \\ 
	& 8/9 & 19.2 & 2.94 \\ 
	& 9/10 & 19.9 & 2.98 \\ 
	\hline
    \end{tabular}
\end{table}


\section{Conclusions}\label{s:conclusions}
We have investigated the TF packing technique, jointly with an advanced processing at the receiver, to improve the spectral efficiency of a nonlinear satellite system employing linear modulations with finite constellations. 
As a first step, through an information-theoretic analysis, we computed the spectral efficiency achievable through this technique showing, with reference to the DVB-S2 specifications, that without an advanced processing at 
the receiver, the potential gains are very limited. On the other hand, a detector which takes into account a memory of only one symbol, and thus with a very limited complexity increase, it is possible to obtain a gain up to 
40\% in terms of spectral efficiency with respect to the conventional use of the current standard. This impressive gain is partly due to optimized carrier spacing of adjacent transponders. Although this assumption 
may not be applicable to all satellite communication systems, the results of this chapter indicates possible new system design directions to further improve the spectral efficiency. 
All these considerations can be extended to other channels and scenarios.
{"config": "arxiv", "file": "1406.5685/cap5.tex"}