TITLE: What is the "standard" way of proving that the transition map is $C^{\infty}$?
QUESTION [2 upvotes]: Sorry if this is a vague question.
I was handed a map between two manifolds, say, $M\to \mathbb{R^n}$. I was indeed able to find an open cover of $M$ and show that there is a homeomorphism between each open set and some open subset of $\mathbb{R^n}$. Proving homeomorphism usually takes me a long time but it's usually doable. But when it comes to showing that each chart is $C^{\infty}$-compatible, in other words, every transition map is smooth, I'm getting nowhere.
The same situation again happened on proving that Grassmannian in the case of $G(2,4)$ has a smooth structure. I identified each open set that covers $G(2,4)$ and proved homeomorphism, but then when it comes to proving the smoothness of transition maps, I get nowhere again. I get the map between two set of matrices and I had no idea how to argue. I think this happens because general theorems on homeomorphism are well-known but smoothness is really different for every situation. But since I am really having trouble proving smoothness, I'd like to know if there is any standard way or theorem of showing that the transition map is smooth. Thank you for your answers.

REPLY [1 votes]: Well, if you're working with $\mathbb{R}^n$, smoothness just means that every component is smooth in all the others (due to some MVT argument, it even suffices to be smooth in every variable separately). And the next trick is that everything you are doing in the Grassmann example is rational (as in: your transition functions are ultimately rational functions), and rational functions are always smooth (where defined).


text:
\begin{document}

\pagestyle{headings}

\renewcommand{\leftmark}{DRAFT: \today}

\maketitle

\begin{abstract}
The pragmatic approach to coded continuous-phase modulation (CPM) is proposed
as a capacity-achieving low-complexity alternative to the serially-concatenated
CPM (SC-CPM) coding scheme.
In this paper, we first perform a selection of the best spectrally-efficient
CPM modulations to be embedded into SC-CPM schemes. Then, we consider the pragmatic
capacity (a.k.a. BICM capacity) of CPM modulations and optimize it through a careful
design of the mapping between input bits and CPM waveforms.
The so obtained schemes are cascaded with an outer serially-concatenated convolutional
code to form a pragmatic coded-modulation system.
The resulting schemes exhibit performance very close to the CPM capacity without
requiring iterations between the outer decoder and the CPM demodulator. As a result,
the receiver exhibits reduced complexity and increased flexibility due to the separation
of the demodulation and decoding functions.
\end{abstract}

\section{Introduction}

Continuous phase modulation (CPM) is a class of bandwidth efficient
modulation schemes~\cite{bib:DigitalPhaseModulation} whose characteristic
of constant envelope makes them robust with respect to the nonlinearities
introduced by the analog baseband and radio-frequency sections of low-cost
transceivers usually found in consumer-type communications equipment.
Good application examples are the second generation GSM cellular system,
and satellite communication systems.

A CPM modulator is a finite-state machine delivering to the channel
a continuous-phase, constant envelope waveform that depends on its input
symbol and internal state.
In~\cite{bib:RimoldiDecomposition} a CPM modulator has been shown to be
decomposable into the cascade of a time-invariant convolutional encoder
(continuous-phase encoder, CPE) operating on a ring of integers, and of
a time-invariant memoryless modulator (MM).
Recently, this decomposition has been exploited by inserting an outer
convolutional encoder whose coded bits enter an interleaver
and then the CPE, thus forming what is known in the literature
as a serially-concatenated convolutional encoder (SCCC)~\cite{bib:Benedetto-Serial}.
In the following, we will call this scheme SC-CPM.

Iterating between the outer encoder and the CPE through the interleaver yields rather good
performance~\cite{bib:MoqvistSCCPM, bib:StuberNarayananTrellisCPM, bib:NarayananMSK},
which should be compared with the capacity of the CPM scheme.
Recently, the authors of~\cite{bib:LoeligerInfoRateMem}
proposed a simulation-based method for the computation of the capacity of channels with memory,
which can be applied to the evaluation of the capacity of the CPM schemes.   A similar
method has been proposed in ~\cite{bib:GeneralCPM}, where it has been applied to
a generalized form of CPM modulation achieving improved spectral efficiency.

In communication systems where the channel conditions can vary significantly with time,
an efficient radio resource management requires the availability at the physical layer of
adaptive coding-modulation, capable of varying its characteristics of bandwidth and
energy efficiency following the channel rate of variation.
This requirement has originated an active research on bit-interleaved (also known as
\emph{pragmatic}) coded modulation \cite{bib:ZehaviPragmatic, bib:ZehaviTrellis, bib:BICM}.
Pragmatic coded modulation consists in cascading a highly performing versatile binary encoder
(typically, a punctured turbo or low-density parity-check code)
with several modulation schemes with increasingly large signal alphabets.
The versatile encoder is capable of varying both its rate and codeword length in a wide range
in order to achieve increasing spectral efficiencies.
An example of the obtainable results has been published in \cite{bib:MHOMS}, which demonstrates
a scheme based on SCCC and linear two-dimensional modulations yielding spectral efficiencies
in a very wide range lying around 1 dB from the Shannon capacity limits.
This paper deals with the extension of the pragmatic approach to CPM modulation (called P-CPM in the following).
This approach does not require iterations between the outer encoder and the
CPE, since the CPM is treated exactly as a linear modulation in a bit-interleaved turbo-trellis
coded modulation.
A nice consequence is that the overall CPE state complexity is not enhanced by the number of
iterations, thus permitting to increase the bandwidth efficiency through the use of a larger
number of CPE states.

In~\cite{bib:BenedettoITA2007} a first attempt to design pragmatic schemes employing
CPM modulation has been presented.
The authors showed that the pragmatic capacity of CPM schemes heavily depends on the
mapping between information bits and CPM signals, and presented a simulation-based mapping
optimization algorithm with some examples.
Further improvements have been presented in~\cite{bib:BenedettoGlobecom2007}, where
a procedure for optimizing such mapping has been proposed.  The optimized CPM modulators
have been embedded into a P-CPM scheme and its performance has been assessed through simulation.
For the considered CPM schemes, performance improvements of more than 2 dB in pragmatic
capacity have been observed.

Previuos literature on the subject includes~\cite{bib:ShaneWeselPCCC-CPM}, where a scheme
consisting of a parallel concatenated turbo code and continuous-phase modulation has been
investigated and a modified encoder has been proposed.
In~\cite{bib:giann05}, CPM modulations have been studied in the
context of multiple antenna systems with layered space-time coding. Reduced-complexity receivers
have been proposed and differential encoding has been introduced in order to obtain an
increased coding gain.

This paper presents a systematic and comprehensive approach to the problem of designing capacity
approaching SC-CPM and P-CPM, through the evaluation of the CPM and P-CPM capacities, the search
for optimal (in the sense of offering the best trade-off between bandwidth efficiency, energy
efficiency and complexity) CPM modulations, their embedding into SC-CPM and P-CPM schemes through the optimization of the mapping between coded bits and CPM waveforms, and a thorough comparison of their error probability performance with respect to capacities. Examples refer to rectangular and raised cosine frequency waveforms, and to three spectral efficiencies deemed important for the applications.

This paper is organized as follows: in Sec.~\ref{sec:system-description} a general
description of the considered coding and modulation systems is given.
Sec.~\ref{sec:compcap} presents the procedure used for the CPM capacity computation
through simulation. In Sec.~\ref{sec:opt} the optimization procedure used to select
the best CPM parameters to be used in SC-CPM schemes is defined.
Sec.~\ref{sec:pragcapopt} presents the procedure used for the optimization of P-CPM
schemes: first, the optimal mapping is derived, the corresponding optimized CPE scheme
is given and then the CPM schemes with best pragmatic capacity are selected.
Sec.~\ref{sec:implementation} shows how the optimized CPM schemes have been embedded
into SC-CPM and P-CPM coded systems.
Finally, Sec.~\ref{sec:results} shows the results obtained using the selected CPM
modulations in SC-CPM and P-CPM schemes.

\section{System description}
\label{sec:system-description}

A CPM modulator is a device with memory that generates continuous-phase,
constant-envelope modulated waveforms

\begin{equation}
x(t) = \sqrt{\frac{2E_s}{T}} e^{j \psi(t)}
\end{equation}

\noindent whose phase

\begin{equation}
\psi(t) = 2 \pi h \sum_{n = -\infty}^\infty{a_n q(t - nT)}
\end{equation}

\noindent depends on the input information symbols $a_n \in \{\pm 1,
\pm 3, \ldots, \pm (M - 1) \}$, where $M = 2^m$ is the size of the
input alphabet. Here, $T$ is the symbol interval, $E_s$ is the energy
per symbol, $h = Q/P$ is the \emph{modulation index} ($Q$ and $P$ are
relatively prime integers), and $q(t)$ is the phase pulse, a
continuous function with the following properties

\[q(t) = \left\{\begin{tabular}{ll}
0           &   $t \leq 0$ \\
$\frac{1}{2}$ & $t \geq LT$
\end{tabular}
\right.\]

\noindent The phase pulse is usually defined as the integral of a
\emph{frequency pulse} $s(t)$

\[q(t) = \int_{-\infty}^t{s(\tau) d \tau}\]

\noindent A CPM scheme is then defined by specifying its
parameters $M$, $h$, $L$ and the frequency pulse $s(t)$.

In this paper, we will consider rectangular (REC) and raised-cosine
(RC) frequency pulses. The REC pulse is defined as

\[s_{\rm REC}(t)=\frac{1}{2LT} [u(t) - u(t - LT)] \]

\noindent and the RC pulse is defined as

\[s_{\rm RC}(t)=\frac{\pi}{4LT} [u(t) - u(t - LT)]
\left[1 - \cos\left(\frac{2 \pi t}{LT}\right)\right] \]

\noindent where $u(t)$ is the unitary step function.

According to the well known Rimoldi decomposition
\cite{bib:RimoldiDecomposition}, the modulator can be represented as
the cascade of a continuous phase encoder (CPE) and a memoryless
modulator (MM) as in Fig.~\ref{fig:cpmenc}. The CPE, in general, is a time-invariant convolutional
encoder operating on a ring of integers.

\begin{figure}[!ht]
\centering
\includegraphics[width=3.5in]{fig/cpmenc.pdf}
\caption{Rimoldi decomposition of the CPM encoder. The block labelled
$\Sigma_P$ is a modulo-$P$ adder.}
\label{fig:cpmenc}
\end{figure}

The serial structure of the Rimoldi decomposition has been exploited by adding an
outer convolutional encoder connected to the CPE trough an interleaver, so as to form a serially concatenated convolutional encoder (SCCC) with interleaver \cite{bib:ett} (see Fig.~\ref{fig:SCCPMcodec}).  The iterative receiver performs decoding iterations
between the inner SISO decoder, which operates on the CPE trellis, and
the outer SISO decoder, which operates on the outer convolutional code
trellis. This way,  very good performance can be achieved, to be compared with the capacity of the CPM scheme. In the following, we will denote this scheme as serially concatenated CPM (SC-CPM)~\cite{bib:MoqvistSCCPM}.

\begin{figure}
\centering
\includegraphics[clip,width=3.5in]{fig/sccpmcodec.pdf}
\caption{Block diagram of a serially concatenated CPM co-decoder.
The block labeled $\cal M$ maps blocks of $m$ bits to $M$-ary CPM symbols.}
\label{fig:SCCPMcodec}
\end{figure}

In this paper, we  also consider  a \emph{pragmatic} approach, which results in a coded
modulation scheme equivalent to a \emph{bit-interleaved} coded
modulation (BICM, see Fig.~\ref{fig:CPMcodec}).    It consists in
cascading a powerful variable-rate binary encoder (a serially-concatenated convolutional code (SCCC) with interleaver  in this paper) with
the CPM modulator. The CPM modulator input is connected to the output
of the channel encoder  that computes the binary sequence
$\mathbf{b} = (b_i, i \in \mathbb{Z})$ from the binary information
sequence $\mathbf{d}$. Then,the coded binary sequence is mapped to the CPM modulator
that generates a corresponding CPM signal sequence $x(t; \mathbf{b})$. We will call this scheme pragmatic CPM (P-CPM).

The channel is an additive white Gaussian noise (AWGN) channel, whose output signal is $y(t; \mathbf{b}) = x(t; \mathbf{b}) + n(t)$, where $n(t)$ is a zero-mean white Gaussian process with two-sided power spectral density $N_0 / 2$.   At the receiver, the CPM soft demodulator and demapper provide the sequence
{\boldmath $\lambda$}~=~$(\lambda_i, i \in \mathbb{Z})$ of
log-likelihood ratios on the information bit sequence to the outer
decoder

\[\lambda_i = \log\frac{P(b_i = 1 | y(t; \mathbf{b}))}{P(b_i = 0 |
y(t; \mathbf{b}))}\]

\noindent Finally, {\boldmath $\lambda$} is used by the outer iterative decoder
to compute the information sequence estimate
$\mathbf{\hat{d}}$.

The main difference between P-CPM and SC-CPM is that no iterations are performed
between the inner CPM demodulator and the outer turbo decoder. Thus, it keeps the nice features of all pragmatic approaches, which merge independent binary codes with higher order modulations without requiring joint optimization.

\begin{figure}
\centering
\includegraphics[clip,width=3.5in]{fig/cpmcodec.pdf}
\caption{Block diagram of an encoded CPM mo-demodulator used with
a pragmatic approach.} \label{fig:CPMcodec}
\end{figure}

Our goal is to design coded modulation schemes based on CPM capable of transmitting
information at rates close to the CPM channel
capacity\footnote{Hereafter, with \emph{CPM channel capacity} we mean
\emph{constrained} capacity with uniform input distribution.}.
Such goal will be pursued by analyzing the P-CPM and SC-CPM schemes
and choosing the best trade-off between the following characteristics:

\begin{itemize}
 \item Spectral efficiency.
 \item Energy efficiency.
 \item Decoding complexity.
\end{itemize}

Spectral efficiency and energy efficiency are two well-known concepts.
The decoding complexity is defined here as the overall number of trellis
edges per information bit visited by the decoding algorithm. This definition depends on the type of
coded modulation being used (i.e., P-CPM or SC-CPM).   For the
P-CPM scheme, we have

\begin{equation}
\label{eq:complexity-pcpm}
\mathcal{Y}_{\rm P-CPM} = \mathcal{Y}_{\rm SCCC} + \mathcal{Y}_{\rm
CPM}
\end{equation}

\noindent where

\[\mathcal{Y}_{\rm CPM} = M N_{s,CPE} \frac{N_o}{m K_o}\]

\noindent and $N_o$  and $K_o$ are, respectively, the outer code
word length and information word length, and $N_{s,CPE} = P
M^{L-1}$ is the number of states of the CPE. The complexity of the
SCCC binary decoder is  $\mathcal{Y}_{\rm SCCC}$ and will be
defined later according to the chosen code (see (\ref{eq:SCCCComplexity})).

For the SC-CPM scheme, we adopt the following definition, which holds
for binary convolutional codes with rate $\frac{K_o}{N_o}$ obtained by
puncturing a rate 1/2 mother code:

\begin{equation}
\label{eq:complexity-sccpm}
\mathcal{Y}_{\rm SC-CPM} = N_{it}\left(2 N_{so} + \mathcal{Y}_{\rm
CPM}\right)
\end{equation}

\noindent where $N_{it}$ is the number of decoding iterations between the
outer SISO and the inner CPE SISO, and $N_{so}$ is the number of states of
the outer convolutional encoder.

In the following, we will optimize the CPM schemes to be embedded into
both SC-CPM and P-CPM starting from their capacity evaluation.

\section{CPM capacity computation}
\label{sec:compcap}

With reference to Figures~\ref{fig:SCCPMcodec} and \ref{fig:CPMcodec}, CPM signals are
infinite-length waveforms.  Consider then a finite observation window
$[-NT,NT]$ of $2N+1$ symbol intervals, and define the channel mutual
information over it:
\[
I({\rm \bf X};{\rm \bf Y})=E_{\mathbf{x}, \mathbf{y}
}\left\{\log_2\frac{p_{\bf Y|X}({\rm \bf y}|{\rm \bf x})}{p_{\bf
Y}({\rm \bf y})}\right\}
\]

\noindent where ${\rm \bf X}$ is a vector of  samples of $x(t)$
that form a sufficient statistic of $x(t)$ in the  interval
$[-NT,NT]$ and ${\rm \bf Y}$ is the correspondent vector of
channel outputs. The set of all possible values for ${\rm \bf X}$
will be denoted $\mathcal X$ throughout the paper. The CPM
capacity can be defined through the limit

\begin{equation}
C_{\mathrm{CPM}}=\lim_{N\rightarrow\infty}\frac{1}{(2N+1)T}I({\rm \bf
X};{\rm \bf Y}) \;\;{\rm [bits/s]}\label{eq:a}
\end{equation}

Using the definition of mutual information $I$ and of the log-likelihood ratio (LLR) $$\lambda({\rm \bf
u},{\rm \bf y} )\triangleq \log_2(p_{\bf Y|X}({{\rm  \bf y}|{\rm \bf
u}))-\log_2(p_{\bf Y|X}({\rm \bf y}|{\rm \bf u}_{\rm ref}))}$$

\noindent as well as the $\max^*$ operator~\cite{bib:ett}

    \[ {\max}^*(a,b) \triangleq  \log_2(2^a+2^b)

    \]
we can transform \refeq{a} into
    \[
    C_{\mathrm{CPM}} =m - \lim_{N\rightarrow\infty}
    \frac{1}{(2N+1)T} E_{\mathbf{x}, \mathbf{y}}\left\{{\max_{\rm \bf u \in \mathcal{X}}}^*
    \lambda({\rm \bf u},{\rm \bf y})-\lambda({\rm \bf x},{\rm \bf y})\right\}.
    \]
where $m$ is the number of bits per CPM symbol.

It has been observed in \cite{bib:LoeligerInfoRateMem} that the
first term inside the average is a by-product of the SISO
algorithm and corresponds to the $\max^*$ of the forward path
metrics $\alpha$ at step $N$ in the CPE decoder
\[
 {\max_{\rm \bf u\in \mathcal{X}}}^* \lambda({\rm \bf u},{\rm \bf
y})={\max_{s}}^*\alpha_N(s),
\]
with the following initialization

\[ \alpha_{-N}(s)=\left\{
    \begin{array}{ll}
        0, & s=0; \\
        -\infty, & s\neq 0.
    \end{array}
\right.
\]
where $s$ runs in the set of trellis states of the CPE.
\noindent Moreover, since the channel is  memoryless, the second term
is
obtained by summing the LLRs of the transmitted waveforms:
\[
\lambda({\rm \bf x},{\rm \bf y})=\sum_{i=-N}^N \lambda(x_i,y_i).
\]

Finally, invoking the ergodic properties of the system, the
ensemble average can be removed leading to:
\[
C_{\mathrm{CPM}} =m - \lim_{N\rightarrow\infty}
  \frac{1}{2N+1} \left({\max_{\bf u}\in \mathcal{X}}^*
  \lambda({\rm \bf u},{\rm \bf y})-\lambda({\rm \bf x},{\rm \bf y})\right).
\]
Thus, the CPM capacity can be estimated through a Monte-Carlo
simulation of the three internal blocks of
Fig.~\ref{fig:CPMcodec}, i.e., the CPM modulator, the channel,
 and the CPM soft demodulator, followed by
a time average.

\section{Optimization procedure}
\label{sec:opt}

We define the complexity of a CPM scheme as the number of edges per
CPM input bit\footnote{Please note that the definition of
$\mathcal{Y}$ differs from $\mathcal{Y}_{\rm CPM}$ in that the former
is the number of CPE trellis edges per CPM input bit, while the latter
is the number of CPE trellis edges per bit at the channel encoder
input.} ${\mathcal Y} = P 2^{mL} / m$,
where the parameters have been defined in Section II. Our aim is to
maximize the CPM capacity measured in bits/s/Hertz versus the
signal-to-noise ratio for a given complexity.   The optimization
algorithm modifies the CPM
parameters ($m$, $P$, and $L$) yielding the given complexity, and
computes the CPM capacity expressed in bits per CPM waveform for a
given signal-to-noise ratio according to the algorithm described in Sec.~\ref{sec:compcap}.
Then, based on a bandwidth definition
of the CPM scheme, it evaluates the rate and chooses the best scheme
for all signal-to-noise ratios of interest.  In the following, we
describe  the optimization algorithm step by step:

\begin{enumerate}
    \item Define the bandwidth $B$ of the CPM system that contains a
given percentage of the total signal power.
    \item Define the CPM symbol rate as $R_s\triangleq 1/T$.
        \item The {\em symbol} signal-to-noise ratio $E_s/N_0$ is
    given by
    \[
    \frac{E_s}{N_0}=\frac{1}{R_s}\frac{P_T}{N_0}
    \] where $P_T$ is the transmitted power.
    \item Compute the capacity (in bits/symbol) of each CPM scheme using the method described in
Section~\ref{sec:compcap} as a function of the
     symbol SNR $C_{\mathrm{CPM}}=C_{\mathrm{CPM}}(E_s/N_0)$.
    \item Evaluate the CPM capacity as
    \begin{equation}\label{eq:Cdef}
        C = R_s
C_{\mathrm{CPM}}\left(\frac{1}{R_s}\frac{P_T}{N_0}\right)\;\;\text{[bits/s/Hz]
}.
    \end{equation}
       \item All CPM schemes with a given complexity ${\mathcal Y}$  are compared
with respect to $C$ for each value of $\frac{P_T}{N_0}$ and the
best is chosen.
        \item Finally, to obtain the normalized plot that
uses the {\em bit} SNR $E_b/N_0$ on the abscissa we use the
relationship
    \begin{equation}\label{eq:ebn0def}
\frac{E_b}{N_0} = \frac{1}{C_{\mathrm{CPM}}}\frac{E_s}{N_0}=\frac{R_s}{C}
\frac{E_s}{N_0} = \frac{1}{C} \frac{P_T}{N_0}.
    \end{equation}
\end{enumerate}

In Fig.~\ref{fig:SpecEffREC} and Fig.~\ref{fig:SpecEffRC}  we show the
capacity versus $E_b/N_0$ for the best CPM schemes with REC and RC frequency
pulses and complexity ranging from 8 to 512. The bandwidth is defined as the one including
99\% of the total signal power. For comparison purposes, in the figure we have
also plotted the capacity of QPSK and 8PSK modulations
with a square root raised cosine shaping filter with roll-off 0.25, and the unconstrained
capacity of the AWGN channel.

In Table~\ref{tab:bestrec99} and Table~\ref{tab:bestrc99} we list for
each SNR  ($E_s/N_0$) and complexity from 8 to 512 the CPM schemes with
REC and RC frequency pulses
achieving the highest capacity for a given signal-to-noise ratio.
Each scheme is characterized by the parameters $E_b/N_0$ as defined in (\ref{eq:ebn0def}),
the capacity $C$ as defined in (\ref{eq:Cdef}), the CPM parameters $m$, $L$, $P$, and finally the
symbol rate $R_s$.

\begin{figure}[ht]
\centering\includegraphics[angle=-90,width=3.5in]{fig/SpecEffREC.pdf}
\caption{Capacity versus $E_b/N_0$ of the best CPM schemes
with rectangular frequency pulse and variable complexity. The
bandwidth is defined at 99 \% of the total power.}
\label{fig:SpecEffREC}
\end{figure}

\begin{figure}[ht]
\centering\includegraphics[angle=-90,width=3.5in]{fig/SpecEffRC.pdf}
\caption{Capacity versus $E_b/N_0$ of the best CPM
schemes with raised-cosine frequency pulse and variable complexity.
The bandwidth is defined at 99 \% of the total power.}
\label{fig:SpecEffRC}
\end{figure}

\begin{table*}[ht]
\centering\includegraphics[angle=-90,width=7in]{fig/tab-rec-cpm.pdf}
\caption{Table of the CPM schemes with best $C$ and rectangular
frequency pulse at 99\% bandwidth. The complexity range is 8 to 512. The highlighted entries
correspond to the selected CPM schemes for target capacities of 1, 1.5 and 2 bits/s/Hz.}
\label{tab:bestrec99}
\end{table*}

\begin{table*}[ht]
\centering\includegraphics[angle=-90,width=7in]{fig/tab-rc-cpm.pdf}
\caption{Table of the CPM schemes with best $C$ and raised-cosine
frequency pulse at 99\% bandwidth. The complexity range is 8 to 512. The highlighted entries
correspond to the selected CPM schemes for target capacities of 1, 1.5 and 2 bits/s/Hz.}
\label{tab:bestrc99}
\end{table*}

\section{Pragmatic capacity of a TCM scheme}
\label{sec:pragcapopt}

With reference to Fig.~\ref{fig:CPMcodec}, let the binary sequence
{\bf b}  be formed by vectors ${\rm \bf B}_n = (B_{n,1}, \dots,
B_{n,m})$ of the $m$ bits entering the CPM modulator at the $n$-th
trellis step. As previously, we consider a length-$(2N+1)$ input
sequence of ${\rm \bf B}_n$.

The scheme of Fig.~\ref{fig:CPMcodec} can be characterized by a
\emph{pragmatic} capacity $C_{\mathrm{P-CPM}}$,
 defined by:
\begin{equation}
C_{\mathrm{P-CPM}} \triangleq
\lim_{N\rightarrow\infty}\frac{1}{2N+1} \sum_{n=-N}^N \sum_{i=1}^m
I(B_{n,i};{\rm \bf Y}), \label{eq:cpcpm}
\end{equation}
which depends on the \emph{mapping} between input binary sequences
and CPM waveforms.

In order to optimize the system performance, it is crucial
to search for the optimal mapping, i.e., the one that maximizes
the pragmatic capacity of a given CPM scheme. To this purpose, we will derive in the following a lower
bound to the pragmatic capacity, which makes explicit the
dependence on the mapping. Before delving deeper into the analysis, though,
we show in Fig.~\ref{fig:cap-comp} a comparison between the optimized P-CPM capacity,
obtained through the algorithm derived in the following, and that obtained by a
straightforward application of the mapping induced by the Rimoldi decomposition.
The curves show that the pragmatic capacity of the optimized
CPE is significantly improved with respect to the non optimized CPE.
For the non optimized CPE, the gap between the CPM capacity and the
pragmatic capacity is roughly 1.5 dB, while the pragmatic capacity of
the optimized CPE is very close to the CPM capacity.
The obtained gain is roughly 1.4 dB.

\begin{figure}[!ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/cap-comp.pdf}
\caption{CPM capacity and pragmatic capacities of the binary REC
scheme with $h = 1/2$ and $L = 3$.}
\label{fig:cap-comp}
\end{figure}

Since
\begin{equation}
I(B_{n,i};{\rm \bf Y}) = 1 - H(B_{n,i}|{\rm \bf Y}), \label{eq:ih}
\end{equation}

\noindent the optimal mapping is the one that yields a minimum of $\sum_{n}
\sum_{i} H(B_{n,i}|{\rm \bf Y})$. Define $\mathcal{X}_{n,i}(b)$,
$b=0,1$, as the set of CPM waveforms whose mapping satisfies
$B_{n,i} = b$. Obviously, $\mathcal{X} = \mathcal{X}_{n,i}(0) \cup
\mathcal{X}_{n,i}(1)$. We have:

\begin{equation} \label{eq:hby}
H(B_{n,i}| \mathbf{Y} )  = \int H(B_{n,i}|\mathbf{Y}=\mathbf{y})
p_{\mathbf{Y}}(\mathbf{y}) d\mathbf{y}.
\end{equation}

Now, $H(B_{n,i}|\mathbf{Y}=\mathbf{y})$ is given explicitly by:

\begin{equation} \label{eq:hby1}
H(B_{n,i}|\mathbf{Y}= \mathbf{y}) = H\left( \frac{\sum_{\mathbf{u}
\in \mathcal{X}_{n,i}(0)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{y} |
\mathbf{u})}{\sum_{\mathbf{u} \in \mathcal{X}} p_{\mathbf{Y} |
\mathbf{X} }(\mathbf{y} | \mathbf{u})}\right),
\end{equation}

\noindent where the binary entropy function $H(p) =-p\log p -(1-p)\log(1-p)$
has been used in the RHS of the above equation.

To find a viable path to mapping optimization, we approximate the
distribution of $\mathbf{Y}$ with its value for infinite SNR, by
making in (\ref{eq:hby}) the following substitution:

\begin{equation}
p_{\mathbf{Y}}(\mathbf{y}) \approx \frac{1}{|\mathcal{X}|}
\sum_{\mathbf{x}
 \in \mathcal{X}} \delta(\mathbf{x} -\mathbf{y}), \label{eq:hsnr}
\end{equation}

\noindent where $|\mathcal{X}|$ is the size of $\mathcal{X}$ and
$\delta(\cdot)$ is Dirac's delta. The above approximation makes
sense because the optimal mapping for large SNR should reasonably
be the same as for infinite SNR.

Substituting \refeq{hsnr} into \refeq{hby}, we obtain:

\begin{equation}
\begin{split}
H(B_{n,i}|\mathbf{Y}) \approx & \frac1{2} \sum_{b=0,1}
\frac{2}{|\mathcal{X}|} \times \\ & \sum_{\mathbf{x} \in \mathcal{X}_{n,i}(b)}
H\left( \frac{\sum_{\mathbf{u} \in \mathcal{X}_{n,i}(0)} p_{\mathbf{Y}
| \mathbf{X}}(\mathbf{x} | \mathbf{u})}{\sum_{\mathbf{u} \in
\mathcal{X}} p_{\mathbf{Y} |
\mathbf{X} }(\mathbf{x} | \mathbf{u})}\right),
\end{split}
\end{equation}

\noindent where the sum w.r.t $\mathbf{x}$ over $\mathcal{X}$ has been split
into separate sums over $\mathcal{X}_{n,i}(0)$ and
$\mathcal{X}_{n,i}(1)$.

By Jensen's inequality and the concavity of the entropy function,
we obtain the upper bound:
\begin{equation} \label{eq:hb0y1}
H(B_{n,i}|\mathbf{Y}) \leq \frac1{2} \sum_{b=0,1} H\left(
\frac{2}{|\mathcal{X}|} \sum_{\mathbf{x}
 \in \mathcal{X}_{n,i}(b)}\frac{\sum_{\mathbf{u} \in
\mathcal{X}_{n,i}(0)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} |
\mathbf{u})}{\sum_{\mathbf{u} \in \mathcal{X}} p_{\mathbf{Y} |
\mathbf{X} }(\mathbf{x} | \mathbf{u})}\right).
\end{equation}

Owing to the ergodic property of the system, it can be shown that
$\sum_{\mathbf{u} \in \mathcal{X}} p_{\mathbf{Y} | \mathbf{X}
}(\mathbf{x} | \mathbf{u})$ is equal to a constant for almost every
transmitted $\mathbf{x}$. Thus:
\begin{equation} \label{eq:hb0y2}
H(B_{n,i}| \mathbf{Y} ) \leq  \frac1{2} \sum_{b=0,1} H\left(
\frac{2\sum_{\mathbf{x}
 \in \mathcal{X}_{n,i}(b)}\sum_{\mathbf{u} \in
\mathcal{X}_{n,i}(0)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} |
\mathbf{u})}{\sum_{\mathbf{x}
 \in \mathcal{X}}\sum_{\mathbf{u} \in
\mathcal{X}} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} | \mathbf{u})}
\right).
\end{equation}

Finally, notice that, since $p_{\mathbf{Y} |
\mathbf{X}}(\mathbf{x} | \mathbf{u}) = p_{\mathbf{Y} |
\mathbf{X}}(\mathbf{u} | \mathbf{x})$ for the AWGN channel, the two
entropies in the RHS of \refeq{hb0y2} have the same arguments, and
thus are equal:
\begin{equation} \label{eq:hb0y3}
H(B_{n,i}| \mathbf{Y} ) \leq  H\left( \frac{2\sum_{\mathbf{x}
 \in \mathcal{X}_{n,i}(0)}\sum_{\mathbf{u} \in
\mathcal{X}_{n,i}(0)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} |
\mathbf{u})}{\sum_{\mathbf{x}
 \in \mathcal{X}}\sum_{\mathbf{u} \in
\mathcal{X}} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} |
\mathbf{u})}\right)
\end{equation}
\noindent or, in a more explicit form:
\begin{equation*} \label{eq:hb0y4}
\begin{split}
H(B_{n,i}| \mathbf{Y} ) \leq - & \frac{2\sum_{\mathbf{x}
 \in \mathcal{X}_{n,i}(0)}\sum_{\mathbf{u} \in
\mathcal{X}_{n,i}(0)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} |
\mathbf{u})}{\sum_{\mathbf{x}
 \in \mathcal{X}}\sum_{\mathbf{u} \in
\mathcal{X}} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} | \mathbf{u})} \times \\
& \log \frac{2\sum_{\mathbf{x}
 \in \mathcal{X}_{n,i}(0)}\sum_{\mathbf{u} \in
\mathcal{X}_{n,i}(0)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} |
\mathbf{u})}{\sum_{\mathbf{x}
 \in \mathcal{X}}\sum_{\mathbf{u} \in
\mathcal{X}} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} | \mathbf{u})}
\\ - & \frac{2\sum_{\mathbf{x}
 \in \mathcal{X}_{n,i}(0)}\sum_{\mathbf{u} \in
\mathcal{X}_{n,i}(1)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} |
\mathbf{u})}{\sum_{\mathbf{x}
 \in \mathcal{X}}\sum_{\mathbf{u} \in
\mathcal{X}} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} | \mathbf{u})} \times \\
& \log \frac{2\sum_{\mathbf{x}
 \in \mathcal{X}_{n,i}(0)}\sum_{\mathbf{u} \in
\mathcal{X}_{n,i}(1)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} |
\mathbf{u})}{\sum_{\mathbf{x}
 \in \mathcal{X}}\sum_{\mathbf{u} \in
\mathcal{X}} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x} | \mathbf{u})}
\end{split}
\end{equation*}

The above equation, substituted in \refeq{ih} and, then, back in
\refeq{cpcpm}, gives the lower bound on the pragmatic capacity
that is the starting point for our optimization.

\subsection{Mapping optimization}

The problem of mapping optimization has been reduced to the minimization of the
upper bound in \refeq{hb0y3} (or \refeq{hb0y4}). A binary entropy
function is minimized if its argument distribution is made as
unbalanced as possible. For large SNR, $ \sum_{\mathbf{x}
 \in \mathcal{X}_{n,i}(0)}\sum_{\mathbf{u} \in
\mathcal{X}_{n,i}(0)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x},
\mathbf{u}) $ will be the dominant term in \refeq{hb0y3}, because it contains the
term with $\mathbf{x}=\mathbf{u}$; thus, it should be made as close
to 1 as possible. Conversely, the complementary term $
\sum_{\mathbf{x}
 \in \mathcal{X}_{n,i}(0)}\sum_{\mathbf{u} \in
\mathcal{X}_{n,i}(1)} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x},
\mathbf{u}), $ should be made as close to 0 as possible, i.e., it
should be minimized. We have come up with the following design
rule for the optimal mapping:
\begin{equation} \label{eq:mapr}
\mathcal{X}_{n,i}(0) = \arg \min_{\substack{\mathcal{X}' \subset
\mathcal{X} \\ |\mathcal{X}'| = |\mathcal{X}|/2}} \sum_{\mathbf{x}
 \in \mathcal{X}'}\sum_{\mathbf{u} \notin
\mathcal{X}'} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x}, \mathbf{u}).
\end{equation}

For practical reasons, the mapping is generated by an $m$-bit
\emph{labelling} of the trellis edges. Let $\mathcal{T}_{n,i}(0)$
($\mathcal{T}_{n,i}(1)$) be the subset of trellis edges at time
$n$ whose label has a 0 (1) in the $i$-th position. Then,
$\mathcal{X}_{n,i}(0)$ is constituted by all CPM waveforms whose
trellis paths pass through an edge belonging to
$\mathcal{T}_{n,i}(0)$, and the design rule in \refeq{mapr} can be
restated in terms of $\mathcal{T}_{n,i}(0)$.

We impose the condition that the trellis labelling must be
\emph{right-resolving}, i.e., edges leaving the same trellis state
have different binary labels. This condition does not force a
trellis labelling that varies with $n$. Moreover, the design
criterion in \refeq{mapr} is also independent of $n$. From these
facts, we deduce that \emph{the optimal mapping is generated by a
time-invariant trellis labelling}. Thus $\mathcal{T}_{n,i}(0) =
\mathcal{T}_{i}(0)$, for every $n$. Instead, the condition of
having a right-resolving trellis implies that
$\mathcal{T}_{1}(0),\dots,\mathcal{T}_{m}(0)$ cannot be chosen
independently, for $m > 1$.

In practice, we approximate the design rule in \refeq{mapr} with
the following suboptimal rule, which derives from the assumption
that, for large SNR, there is one dominant term in the sum of
\refeq{mapr}:
\begin{equation} \label{eq:mapr1}
\mathcal{X}_{n,i}(0) = \arg \min_{\mathcal{X}' \subset
\mathcal{X}} \left\{\max_{\mathbf{x}
 \in \mathcal{X}'}\max_{\mathbf{u} \notin
\mathcal{X}'} p_{\mathbf{Y} | \mathbf{X}}(\mathbf{x},
\mathbf{u})\right\}.
\end{equation}
Based on \refeq{mapr1}, our mapping optimization procedure
consists then of the following steps:
\begin{itemize}
\item For every pair of trellis edges, we apply the BCJR algorithm
to compute ${\max_{\mathbf{u}}} {\max_{\mathbf{x}}} p_{\mathbf{Y}
| \mathbf{X}}(\mathbf{x}, \mathbf{u})$ over all pairs of paths
passing through the given pair of edges at time zero. To do this
in general, we have to extend the trellis both at the left
(negative time instants) and at the right (positive time instants)
of the zeroth section. After a few trellis steps, the path metrics
reach a steady-state value, so there is no need to proceed
further.

\item The previous step yields a metric for each pair of edges.
Since we want to minimize the expression between braces in
\refeq{mapr1}, we partition the edges into $M = 2^m$ equal-size
clusters, corresponding to the $M$ different $m$-bit labels, in
such a way that pairs of edges with the highest metric are all
clustered together. In doing this, we force pairs of edges with
the same starting state into different clusters, to allow for a
right-resolving labelling.

\item We map the $M = 2^m$ clusters to $m$-tuples of bits
according to a Gray mapping. More precisely, we define the
distance between two clusters as the minimum distance between CPM
waveforms passing at time zero through edges belonging to the two
clusters. Cluster pairs with the smallest distance will be
associated to binary labels with Hamming distance 1.
\end{itemize}

The highest metric corresponds to edge pairs that belong to the
same trellis section of a minimum-distance error event. It may be
questionable whether it is possible to cluster \emph{all} pairs of
edges with that property, apart from those with the same starting
state, and whether this gives a \emph{unique} clustering. In the
next subsection, we will completely answer these questions for
$M=2$, and give the analytical expression of the optimal mapping,
provided that some necessary conditions are met.

\subsection{The optimal mapping for $M=2$}
\label{sec:pragcap}

As described in the previous subsection, the algorithm for mapping
optimization puts into the same cluster all edge pairs that belong
to the same trellis section of a minimum-distance error event. For
labelling purposes, we would like to obtain in this way exactly
$M$ clusters, but this happens only at certain conditions. In this
subsection, we derive such conditions for \emph{binary} CPM
schemes.

Let us consider a CPM scheme with $M=2$, impulse length $L$ and
modulation index $h=Q/P$. Trellis edges will be denoted hereafter
through the couple $(\mathbf{\alpha}, \beta)$, where
$\mathbf{\alpha}$ is a length-$L$ binary vector, including the
correlative state and the input symbol, and $\beta \in \{0,\dots,
P-1\}$ represents the phase state. Let $x_1(t)$ and $x_2(t)$ be
two CPM waveforms, corresponding to input symbol sequences
$\mathbf{a}_1$ and $\mathbf{a}_2$, respectively. It is well known
that the Euclidean distance between these two CPM waveforms only
depends on the \emph{difference} sequence $\mathbf{b} \triangleq
\mathbf{a}_1-\mathbf{a}_2$, with elements belonging to
$\{-1,0,1\}$.

Now, consider a given difference sequence $\mathbf{b}$ and define
a graph $\mathcal{G}(\mathbf{b}) = (\mathcal{V},
\mathcal{E}(\mathbf{b}))$, where:
\begin{itemize}
\item the vertex set $\mathcal{V}$ is in one-to-one correspondence
to the set of trellis edges for the considered CPM scheme.

\item the edge set $\mathcal{E}(\mathbf{b})$ is constructed in the
following way: two vertices are connected by an edge if and only
if the corresponding trellis edges have different starting states
and belong to the same trellis section of an error event generated
by $\mathbf{b}$.
\end{itemize}

Let $\mathcal{C}_0, \dots,
\mathcal{C}_{\mathcal{N}(\mathbf{b})-1}$ be the
$\mathcal{N}(\mathbf{b})$ connected components of the graph
$\mathcal{G}(\mathbf{b})$. The following theorem gives the
important properties of $\mathcal{C}_0, \dots,
\mathcal{C}_{\mathcal{N}(\mathbf{b})-1}$.

\begin{theorem} \label{th:1}
Consider a CPM scheme with $M=2$, impulse length $L$ and
modulation index $h=Q/P$. Given a difference sequence
$\mathbf{b}$, with length $\Delta(\mathbf{b})$, the following
facts about the graph $\mathcal{G}(\mathbf{b})$ defined above
hold:
\begin{enumerate}
\item If $\Delta(\mathbf{b}) = 2$, then $\mathcal{N}(\mathbf{b}) =
P$, otherwise $\mathcal{N}(\mathbf{b}) = 1$.

\item If $\mathcal{N}(\mathbf{b}) = P$, the connected component
$\mathcal{C}_i$, $i=0,\dots,P-1$ is the subgraph induced by the
subset of vertices\footnote{The notation $l_P$ means ``$l$ modulo
$P$''.}:
\begin{equation}
\mathcal{V}_i = \left\{\left(\mathbf{\alpha}, (i- Q
w_H(\mathbf{\alpha}))_P \right) : \mathbf{\alpha} \in \{0,1\}^L
\right\},
\end{equation}
independently of $\mathbf{b}$. All $\mathcal{C}_i$'s have size
$2^L$.

\end{enumerate}
\end{theorem}

\proof See Appendix \ref{sec:App1}.
\endproof

Suppose now that $\mathbf{b}$ gives the minimum Euclidean distance
of the CPM scheme. We want to partition the edges into two
clusters, namely, $\mathcal{T}(0)$ and $\mathcal{T}(1)$ (edges
with a label 0 and edges with a label 1, respectively). To meet
all constraints given by the graph $\mathcal{G}(\mathbf{b})$,
every subset $\mathcal{V}_i$ should be entirely contained into one
of the two clusters. This is clearly not possible if
$\Delta(\mathbf{b}) \neq 2$, because in that case
$\mathcal{V}_1=\mathcal{V}$.

If $\Delta(\mathbf{b}) = 2$, instead, we can place exactly $P/2$
subsets into $\mathcal{T}(0)$ and the other $P/2$ into
$\mathcal{T}(1)$. (Notice that the sizes of $\mathcal{T}(0)$ and
of $\mathcal{T}(1)$ both must be equal to $P2^{L-1}$.) This can be
done only if $P$ is even, otherwise, one of the $\mathcal{V}_i$'s
must be split in two. Thus, we have proved the following
proposition:
\begin{proposition} \label{prop:2}
Consider a CPM scheme with $M=2$, impulse length $L$ and
modulation index $h=Q/P$. We can fully perform the algorithm of
mapping optimization described in the previous section if and only
if:
\begin{itemize}
\item Every difference sequence $\mathbf{b}$ giving the minimum
Euclidean distance of the scheme has length two, and

\item $P$ is even.
\end{itemize}
\end{proposition}

If the conditions of Prop. \ref{prop:2} are satisfied, then, the
optimal mapping is uniquely determined by the condition of
right-resolving labelling. In fact, the two trellis edges leaving
a given trellis state will belong to $\mathcal{V}_i$ and
$\mathcal{V}_{(i+Q)_P}$, for some $i$. Thus, for the labelling to
be right-resolving, it must be:
\begin{equation}
\mathcal{T}(0) = \mathcal{V}_0 \cup \mathcal{V}_{(2Q)_P} \cup
\dots \cup \mathcal{V}_{((P-2)Q)_P}
\end{equation}
and
\begin{equation}
\mathcal{T}(1) = \mathcal{V}_{Q_P} \cup \mathcal{V}_{(3Q)_P} \cup
\dots \cup \mathcal{V}_{((P-1)Q)_P},
\end{equation}
or vice versa. In this way, all CPM waveform pairs at a minimum
distance between each other will be associated to binary input
sequence pairs with Hamming distance equal to 1.

If the conditions of Prop. \ref{prop:2} are not satisfied, then
the mapping optimization algorithm of the previous section cannot
be fully performed. It means that some CPM waveform pairs with
minimum distance will be associated to binary input sequence pairs
with Hamming distance larger than 1. A reasonable approach in this
case is to try to minimize the number of such CPM waveform pairs.

\subsection{The optimal mapping for $M>2$}

In non-binary CPM schemes, things become more involved. In general,
we have not found necessary and sufficient conditions to fully
perform the algorithm of mapping optimization.

Here, we give a generalization of Theorem \ref{th:1}, which
however deals only with a subset of possible difference sequences.
\begin{theorem} \label{th:2}
Consider a CPM scheme with $M>2$, impulse length $L$ and
modulation index $h=Q/P$. Let
$\mathbf{b}=(b_1,\dots,b_{\Delta(\mathbf{b})})$  be a difference
sequence  with length $\Delta(\mathbf{b})$ and elements in
$\{0,\pm 1, \dots, \pm (M-1)\}$. Let $b_{\Delta(\mathbf{b})} = \pm
1$. The following facts about the graph $\mathcal{G}(\mathbf{b})$
defined above hold:
\begin{enumerate}
\item If $\Delta(\mathbf{b}) = 2$, then $\mathcal{N}(\mathbf{b}) =
P$, otherwise $\mathcal{N}(\mathbf{b}) = 1$.

\item If $\mathcal{N}(\mathbf{b}) = P$, the connected component
$\mathcal{C}_p$, $p=0,\dots,P-1$ is the subgraph induced by the
subset of vertices
\begin{equation} \label{eq:Vi}
\begin{split}
\mathcal{V}_p = & \left\{\left(\mathbf{\alpha}, \left(p- Q
\sum_{l=1}^L a_l\right)_P \right) : \right. \\
& \left. \mathbf{\alpha}
=(a_1,\dots,a_L) \in \{0,\dots,M-1\}^L \right\},
\end{split}
\end{equation}
independently of $\mathbf{b}$. All $\mathcal{C}_p$'s have size
$M^L$.

\end{enumerate}
\end{theorem}

\proof See Appendix \ref{sec:App2}.
\endproof

Following the same arguments of the previous subsection, we thus
obtain the following proposition, which gives only sufficient (not
necessary) conditions to fully perform the mapping optimization
algorithm.

\begin{proposition} \label{prop:4}
Consider a CPM scheme with $M>2$, impulse length $L$ and
modulation index $h=Q/P$. We can fully perform the algorithm of
mapping optimization described in the previous section if:
\begin{itemize}
\item Every difference sequence $\mathbf{b}$ giving the minimum
Euclidean distance of the scheme has length two and $b_2 = \pm 1$,
and

\item $P$ is a multiple of $M$.
\end{itemize}
\end{proposition}

If the conditions of Prop. \ref{prop:4} are satisfied, then, by
constraining the  resulting labelling to be right-resolving, we
obtain the following $M$ clusters:
\begin{equation} \label{eq:Ti}
\widetilde{\mathcal{T}}(i) = \mathcal{V}_{(iQ)_P} \cup
\mathcal{V}_{((M+i)Q)_P} \cup \dots \cup
\mathcal{V}_{((P-M+i)Q)_P},
\end{equation}
for $i=0,\dots,M-1$.

Finally, the clusters $\widetilde{\mathcal{T}}(i)$'s are mapped to
the $M$ binary labels according to Gray mapping.

In the cases that are not covered by the hypotheses of Theorem
\ref{th:2}, the algorithm may or may not be fully performed
depending on the scheme parameters and on the difference sequences
$\mathbf{b}$ giving the minimum Euclidean distance. If the
algorithm can be fully performed, in general, the $M$ clusters may
not be as in \refeq{Ti}.

\subsection{The optimized CPE}

Starting from (\ref{eq:Vi}), where we define

\[
a_{L+1} = \left(p - Q \sum_{l=1}^L{a_l}\right)_P
\]

\noindent we derive the input symbol $a_1$ to the CPE as the solution
to the following equation

\[
(Q a_1)_P = \left(p - a_{L+1} - Q \sum_{l=2}^L{a_l}\right)_P.
\]

\noindent Since $P$ and $Q$ are relatively prime, such solution is unique.
Moreover, from (\ref{eq:Vi}) we have $a_1 \in \{0, \ldots, M - 1\}$ and,
since also $M$ and $Q$ are relatively prime,  $(Q a_1)_M$ entails a unique
solution for $a_1$.
As a result, we can write

\begin{equation}\label{eq:Qa1}
(Q a_1)_M = \left(p - a_{L+1} - Q \sum_{l=2}^L{a_l}\right)_M.
\end{equation}

Clustering as in (\ref{eq:Ti}) can be performed observing that the vertex subset indices
in each cluster can be written as $p = ((kM + i) Q)_P$.   Inserting this in (\ref{eq:Qa1})
and recalling that $P$ is a multiple of $M$ we obtain

\begin{equation*}\label{eq:Qa1-2}
\begin{split}
(Q a_1)_M & = \left(iQ + k M Q - a_{L+1} - Q \sum_{l=2}^L{a_l}\right)_M \\
& = \left(iQ - a_{L+1} - Q \sum_{l=2}^L{a_l}\right)_M.
\end{split}
\end{equation*}

\noindent where $i$ is the cluster label. Applying the distributive property of multiplication
over addition in the ring $\mathbb{Z}_M$  we obtain

\begin{equation}
a_1 = \left(i - (a_{L+1}Q^{-1})_M - \sum_{l=2}^L{a_l}\right)_M.
\label{eq:a1f}
\end{equation}

\noindent Finally, the binary label $\bf B$ is obtained as
$\mathbf{B} = G(i)$, where $G$ is the Gray function defined in~\cite[ch.20]{bib:nr}.
Fig.~\ref{fig:cpmencopt} shows the scheme
of the optimized CPM encoder.

\begin{figure}[!ht]
\centering
\includegraphics[width=3.5in]{fig/cpmencoptq.pdf}
\caption{The optimized CPM encoder. The block labelled
$\Sigma_P$ (resp. $\Sigma_M$) is a modulo-$P$ (resp. modulo-$M$) adder.
The block labelled $G^{-1}$ is the inverse of the Gray function defined in~\cite[ch.20]{bib:nr}.}
\label{fig:cpmencopt}
\end{figure}

Adopting the optimized CPE, i.e., starting from the CPE structure of Fig.~\ref{fig:cpmencopt},
a procedure similar to the one described in Sec.~\ref{sec:opt} has been
performed in order to select the CPM schemes with best $C_{\rm P-CPM}$.
Results are shown in Tab.~\ref{tab:bestrec99p} and Tab.~\ref{tab:bestrc99p}.

\begin{table*}[ht]
\centering\includegraphics[angle=-90,width=7in]{fig/tab-rec-pcpm.pdf}
\caption{Table of the P-CPM schemes with best $C_{\rm P-CPM}$ and
rectangular frequency pulse at 99\% bandwidth. The
complexity range is 8 to 512. The highlighted entries
correspond to the selected CPM schemes for target capacities of 1, 1.5 and 2 bits/s/Hz.}
\label{tab:bestrec99p}
\end{table*}

\begin{table*}[ht]
\centering\includegraphics[angle=-90,width=7in]{fig/tab-rc-pcpm.pdf}
\caption{Table of the P-CPM schemes with best $C_{\rm P-CPM}$ and
raised-cosine frequency pulse at 99\% bandwidth. The complexity range
is 8 to 512. The highlighted entries
correspond to the selected CPM schemes for target capacities of 1, 1.5 and 2 bits/s/Hz.}
\label{tab:bestrc99p}
\end{table*}

A comparison of Tab.~\ref{tab:bestrec99p} with Tab.~\ref{tab:bestrec99} and
Tab.~\ref{tab:bestrc99p} with Tab.~\ref{tab:bestrc99} highlights that, in some
cases, the CPM schemes with best $C_{\rm CPM}$ feature the best  $C_{\rm P-CPM}$,
as is the case, e.g., for REC pulses with complexity 64 and capacity 1.5 bits/s/Hz
and for RC pulses with complexity 64 and capacity 1.5 bits/s/Hz and 2 bits/s/Hz.
In some other cases, the selected schemes do not match.

We observed that, in many binary CPM schemes, the pragmatic capacity resulting from the
CPE optimization approaches the CPM capacity closer than for non binary schemes.
As a result, even when featuring a worse CPM capacity, these schemes may exhibit a larger
pragmatic capacity.
This is the case, e.g., of REC pulses with complexity 64 and capacity 1 bits/s/Hz, where
the scheme with best $C_{\rm CPM}$ is quaternary and the scheme with best $C_{\rm P-CPM}$
is binary.

The optimal mapping described above results in a CPM scheme with finite $d_{E1}$,
the minimum Euclidean distance at the CPM output corresponding to an input Hamming
distance 1. To prove this, we use the polynomial notation with dummy variable $D$ for
symbol sequences and we observe from Fig.~\ref{fig:cpmencopt} that

\begin{equation}
a_{i+1}(D) = D a_i(D), \quad i = 1, \ldots , L - 1
\label{eq:aip1}
\end{equation}

\noindent and

\begin{equation}
a_{L+1}(D) = \left(a_L(D) \frac{D}{1 - D}\right)_P
\label{eq:alp1}
\end{equation}

\noindent If $Q = 1$, from (\ref{eq:alp1}), (\ref{eq:aip1}) and (\ref{eq:a1f}), we obtain

\[a_1(D) = \left(i(D) (1 - D)\right)_M.\]

\noindent Therefore $a_1(D)$ can be obtained by feedforward encoding
of $i(D)$.   From (\ref{eq:aip1}) it is straightforward to
deduct that all $a_i(D)$, $i = 1, \ldots, L$ have the same property.

As for symbol $a_{L+1}(D)$, from (\ref{eq:alp1}) we obtain

\begin{equation}
a_{L+1}(D) = \left(\frac{D^L a_1(D)}{1 - D}\right)_P = \left(\frac{D^L \left(i(D) (1 - D)\right)_M}{1 - D}\right)_P
\label{eq:alp1-1}
\end{equation}

\noindent If $M = z P, z \in \mathbb{N}$, this clearly proves that symbol $a_{L+1}(D)$ can be
obtained by feedforward encoding $i(D)$.
When $P = w M, w \in \mathbb{N}$, it is possible to rewrite (\ref{eq:alp1-1}) as

\begin{eqnarray}
\nonumber a_{L+1}(D) & = & \left(\frac{D^L \left(i(D) (1 - D)\right)_M}{1 - D}\right)_M \\
\nonumber & + & M \left\lfloor\frac{D^L \left(i(D) (1 - D)\right)_M}{M (1 - D)}\right\rfloor_{P/M} \\
\nonumber & = & \left(D^L i(D)\right)_M \\
& + & M \left\lfloor\frac{D^L \left(i(D) (1 - D)\right)_M}{M (1 - D)}\right\rfloor_{P/M}
\label{eq:alp1-2}
\end{eqnarray}

\noindent When the last term of (\ref{eq:alp1-2}) is zero, $a_{L+1}(D)$ can be obtained by
feedforward encoding from $i(D)$, therefore the encoder is feedforward.
For \emph{difference} sequences $i(D)$ with length 1 and symbols in $\{-1, +1\}$ this condition is met.
Moreover, due to Gray mapping, the corresponding pairs of binary input sequences have unitary
Hamming distance, hence the resulting encoder has finite $d_{E1}$.

It is well known that in serially concatenated schemes, in order to take advantage
of the interleaver gain~\cite{bib:Benedetto-Unveiling}, the inner constituent encoder
or modulator must have infinite $d_{E1}$~\cite{bib:ett}.
For this reason, when used in SC-CPM schemes, the optimal mapping leads to poor
performance and hence a recursive CPE must be employed.
We have thus used the mapping induced by the Rimoldi decomposition, which
has this property.

\section{Implementation of the optimized coded CPM schemes}
\label{sec:implementation}

Using the procedures described in the previous sections, SC-CPM and P-CPM schemes have
been designed with target capacities of $C_T = 1$ bits/s/Hz, $C_T = 1.5$ bits/s/Hz and
$C_T = 2$ bits/s/Hz.

\subsection{Selection of the modulation schemes}

Fig.~\ref{fig:PowerEffREC} and Fig.~\ref{fig:PowerEffRC} show the $E_b/N_0$
values plotted against the CPM complexity for different target capacities.
Continuous curves refer to CPM capacity and dashed curves refer to pragmatic
capacity.

For REC schemes (Fig.~\ref{fig:PowerEffREC}) we observe that increasing the
complexity beyond 64 yields a negligible gain both in CPM capacity and in pragmatic
capacity. Hence, CPM schemes with REC frequency pulse and maximum
complexity 64 have been considered.

For RC schemes (Fig.~\ref{fig:PowerEffRC}), increasing the
complexity beyond 64 yields a significant gain both in CPM capacity and in pragmatic
capacity at 1.5 bits/s/Hz and 2 bits/s/Hz, hence CPM schemes with higher complexities
should be chosen. However, in SC-CPM receivers, the increased complexity would be
enhanced by the iterative process, resulting in an impractical solution.

\begin{figure}[ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/PowerEffREC.pdf}
\caption{Energy efficiency of REC CPM schemes.}
\label{fig:PowerEffREC}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/PowerEffRC.pdf}
\caption{Energy efficiency of RC CPM schemes.}
\label{fig:PowerEffRC}
\end{figure}

\subsection{Serially-concatenated schemes}

A serially concatenated CPM scheme consists of the cascade of an outer convolutional
encoder connected to the CPM modulator through an interleaver.

As outer encoder, a 4-state, rate 1/2 systematic recursive convolutional encoder has been chosen. It is shown in Fig.~\ref{fig:convenc}. The choice has been dictated by a trade-off between performance and complexity. The encoder output can be punctured to achieve rates up to 3/4 while keeping a free distance larger than 2.

\begin{figure}
\centering
\includegraphics[clip,width=3in]{fig/convenc.pdf}
\caption{Convolutional encoder used in SC-CPM schemes.}
\label{fig:convenc}
\end{figure}

In order to obtain rates larger than 1/2, the output of the convolutional
encoder is punctured according to a rate-matching algorithm that selects
all systematic bits and some coded bits.  The puncturing rate is

\[
R_P = \frac{N_O}{N_I} \geq 1
\]

\noindent where $N_O$ and $N_I$ are two integers ($N_I \leq N_O \leq 2 N_I$).
The puncturing algorithm selects $N_I$ coded bits from each block of $N_O$ with
the following constraint: all systematic bits are selected; some coded bits
are selected to achieve the desired rate; the other bits are punctured.
The resulting outer code rate is

\[
R_{CC} = \frac{1}{2}\frac{N_O}{N_I}
\]

The interleaver that connects the punctured outer encoder to the inner CPM modulator
is a spread interleaver.

The design of a SC-CPM scheme consists in the choice of the CPM scheme and of the
outer encoder based on the desired capacity and on design constraints (e.g., the
maximum affordable complexity of the CPM scheme).

\begin{example}
We describe the design of a SC-CPM scheme with target capacity
$C_T = 1.5$ bits/s/Hz, REC frequency pulse and maximum CPM complexity of 64 edges per CPM
input bit.
As for SC-CPM schemes, the parameters that maximize $C_{\rm CPM}$ must be chosen,
from Tab.~\ref{tab:bestrec99}.
Moreover, since the maximum CPM complexity is 64, we must restrict the search to the
column set labelled 64.
Next, in the column labelled $C$, we look for the value closest to the target capacity.
We find that the capacity value closest to $C_T$ is 1.51 and the
corresponding CPM parameters are $m = 2$, $h = 1/5$, $L = 2$.
Such scheme exhibits a symbol rate $R_s = 1.18$ and a corresponding bit rate $R_b = m R_s = 2.36$.
The scheme of the CPM modulator corresponding to these parameters is shown in Fig.~\ref{fig:cpmenc-2-5-2}.

\begin{figure}[ht]
\centering
\includegraphics[width=3.5in]{fig/cpmenc-2-5-2.pdf}
\caption{Scheme of the CPM modulator with $m = 2$, $h = 1/5$, $L = 2$.}
\label{fig:cpmenc-2-5-2}
\end{figure}

In order to obtain a target capacity $C_T = 1.5$ bits/s/Hz, the outer code rate must be set to

\[
R_{CC} = \frac{N_O}{2 N_I} = \frac{C_T}{R_b} = 0.636
\]

\noindent therefore

\[
\frac{N_O}{N_I} = \frac{2 C_T}{R_b} = 1.2712 \simeq \frac{75}{59}.
\]

\noindent The final step consists in choosing two integers for $N_O$ and $N_I$ in order
to approximate with sufficient precision this ratio.

\flushright \QEDopen
\end{example}

The resulting design parameters for all considered target capacities are shown in Tab.~\ref{tab:c-cpm-designs}.

\subsection{Pragmatic schemes}

A pragmatic CPM scheme consists of the cascade of a SCCC encoder and a CPM modulator.

The chosen encoder is the SCCC encoder described in~\cite{bib:MHOMS}.
The constituent convolutional encoders used in such scheme are the 4-state convolutional
encoders shown in Fig.~\ref{fig:convenc}.
The outer convolutional encoder is punctured to a rate 2/3.

The inner encoder is punctured using a carefully designed rate-matching algorithm
that selects all the systematic bits and some coded bits in order to achieve the desired
overall rate.

The interleaver that connects the punctured outer encoder to the inner encoder is a
spread interleaver.

\begin{example}
We show how to design  a P-CPM scheme with target capacity
of 2 bits/s/Hz, RC frequency pulse and maximum CPM complexity of 64 edges per CPM
input bit.
For P-CPM schemes, the inner CPM modulation parameters that maximize $C_{\rm P-CPM}$
must be chosen. Therefore, Tab.~\ref{tab:bestrc99p} must be considered. Moreover, since
the maximum CPM complexity is 64, we must restrict the search to the column set labelled 64.
Next, in column labelled $C$ we look for the value closest to the target capacity.
We find that the capacity value closest to $C_T$ is 2.00 and the
corresponding CPM parameters are $m = 2$, $h = 1/8$, $L = 2$.
Such scheme exhibits a symbol rate $R_s = 1.18$ and a corresponding bit rate
$R_b = m R_s = 2.36$.
The scheme of the CPM modulator corresponding to these parameters is shown in Fig.~\ref{fig:cpmencoptq-2-8-2}.

\begin{figure}[ht]
\centering
\includegraphics[width=3.5in]{fig/cpmencoptq-2-8-2.pdf}
\caption{Scheme of the CPM modulator with $m = 2$, $h = 1/8$, $L = 2$.}
\label{fig:cpmencoptq-2-8-2}
\end{figure}

In order to obtain a target capacity $C_T = 2$ bits/s/Hz, the outer code rate must be

\[
R_{\rm SCCC} = \frac{C_T}{R_b} = 0.847.
\]

\flushright \QEDopen
\end{example}

Tab.~\ref{tab:c-cpm-designs} summarizes the parameters of the designed schemes.

\begin{table}
\centering
\begin{tabular}{r|c||cc|ccc||c|ccc}
    & \multirow{2}{*}{$C_T$} & \multicolumn{5}{c||}{SC-CPM}  &  \multicolumn{4}{c}{P-CPM}   \\
                     &  & $N_O$ & $N_I$ & $m$ & $P$ & $L$ & $R_{\rm SCCC}$ & $m$ & $P$ & $L$      \\
\hline \hline
\multirow{3}{*}{\begin{sideways}REC\end{sideways}} &   1   &  25   &   24  &  2  &  4  &  2  &       0.752    &  1  &  2  &  3  \\
                     &   1.5 &  75   &   59  &  2  &  5  &  2  &       0.781    &  2  &  4  &  2  \\
                     &   2   &  40   &   27  &  2  &  6  &  2  &       0.735    &  1  &  4  &  4  \\
\hline
\multirow{3}{*}{\begin{sideways}RC\end{sideways}}  &   1   &    5  &    4  &  2  &  4  &  2  &       0.625    &  2  &  4  &  2  \\
                     &   1.5 &  150  &   91  &  2  &  5  &  2  &       0.938    &  2  &  4  &  2  \\
                     &   2   &  100  &   59  &  2  &  8  &  2  &       0.847    &  2  &  8  &  2  \\
\end{tabular}
\caption{Parameters of the optimized SC-CPM and P-CPM schemes yielding 1.0, 1.5 and 2.0 bits/s/Hz.}
\label{tab:c-cpm-designs}
\end{table}

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

The performance of the designed coding schemes reported in Tab.~\ref{tab:c-cpm-designs}
has been assessed through simulation.  For each scheme, the CPM capacity and the pragmatic
capacity have been estimated. The obtained results are shown in
Fig.~\ref{fig:C64se-REC} for REC frequency pulses and
Fig.~\ref{fig:C64se-RC} for RC frequency pulses.

\begin{figure}[ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/C64se-REC.pdf}
\caption{CPM capacity and pragmatic capacity of
the selected CPM schemes with complexity $\mathcal{Y} \leq 64$. The
frequency pulse is REC.}
\label{fig:C64se-REC}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/C64se-RC.pdf}
\caption{CPM capacity and pragmatic capacity of
the selected CPM schemes with complexity $\mathcal{Y} \leq 64$. The
frequency pulse is RC.}
\label{fig:C64se-RC}
\end{figure}

In both the P-CPM and the SC-CPM case, an outer information word
length of $K = 9840$ bits has been considered and the code rates have
been set according to Tab.~\ref{tab:c-cpm-designs}.
The number of decoding iterations has been set to 10.

The information rates $I_{\rm SC-CPM}$ (resp. $I_{\rm P-CPM}$) of the SC-CPM (resp. P-CPM)
schemes has been evaluated through simulation by computing the mutual information between
the information bits at the input of the channel encoder and the soft outputs
of the channel decoder.
Fig.~\ref{fig:C64se-1-REC}, Fig.~\ref{fig:C64se-15-REC} and Fig.~\ref{fig:C64se-2-REC}
show the obtained information rates for REC frequency pulse and complexity
${\cal Y} \leq 64$.   We observe that, for a capacity of 1 bits/s/Hz,
the pragmatic scheme achieves the desired information rate
at a lower $E_b/N_0$ than the SC-CPM scheme, and at 1 dB from the pragmatic capacity.
For capacities of 1.5 bits/s/Hz and 2 bits/s/Hz, the SC-CPM scheme
achieves the desired information rate at a lower $E_b/N_0$ than the
P-CPM scheme.

The best SC-CPM and P-CPM for the chosen spectral efficiencies have been simulated over
an AWGN channel so as to obtain accurate estimates of the bit and frame error
probabilities at very low values. The results are shown in Fig.~\ref{fig:C64er-REC}.
We observe that, at 1 bits/s/Hz, the P-CPM scheme, while featuring a lower complexity
(see Tab.~\ref{tab:complexity}), exhibits a 0.5 dB gain over the SC-CPM scheme at
FER = $10^{-4}$.   At higher spectral efficiencies, the SC-CPM scheme exhibits a gain of more than
1 dB over the P-CPM scheme.

Fig.~\ref{fig:C64er-RC} shows similar results for RC frequency pulses.
The SC-CPM schemes feature a 0.3 dB gain over the P-CPM schemes at 1 bits/s/Hz and
2 bits/s/Hz, while at 1.5 bits/s/Hz the gain of the SC-CPM scheme is about 1 dB.
In both frequency pulses and all spectral efficiencies, no error floors have been observed.

\begin{figure}[!ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/C64se-1-REC.pdf}
\caption{Information rates of P-CPM and SC-CPM coded systems at 1
bits/s/Hz. ${\cal Y} = 64$, REC frequency pulse.}
\label{fig:C64se-1-REC}
\end{figure}

\begin{figure}[!ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/C64se-15-REC.pdf}
\caption{Information rates of P-CPM and SC-CPM coded systems at 1.5
bits/s/Hz. ${\cal Y} = 64$, REC frequency pulse.}
\label{fig:C64se-15-REC}
\end{figure}

\begin{figure}[!ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/C64se-2-REC.pdf}
\caption{Information rates of P-CPM and SC-CPM coded systems at 2
bits/s/Hz. ${\cal Y} = 64$, REC frequency pulse.}
\label{fig:C64se-2-REC}
\end{figure}

\begin{figure}[!ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/C64er-REC.pdf}
\caption{Error rates of P-CPM (curves without markers) and SC-CPM
(curves with markers) coded systems. ${\cal Y} = 64$, REC frequency
pulse.}
\label{fig:C64er-REC}
\end{figure}

\begin{figure}[!ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/C64er-RC.pdf}
\caption{Error rates of P-CPM (curves without markers) and SC-CPM
(curves with markers) coded systems. ${\cal Y} = 64$, RC frequency
pulse.}
\label{fig:C64er-RC}
\end{figure}

A comparison of REC and RC pragmatic schemes is shown in Fig.~\ref{fig:C64er-RECvsRC}.
We observe that the RC scheme performs better than REC scheme at 1 bits/s/Hz and 1.5 bits/s/Hz.
At 2 bits/s/Hz, the hierarchy is reversed.
At 2 bits/s/Hz, REC schemes feature a gain of roughly 0.6 dB over the RC schemes at FER = $10^{-4}$.

\begin{figure}[!ht]
\centering
\includegraphics[angle=270,width=3.5in]{fig/C64er-RECvsRC.pdf}
\caption{Comparison of REC (curves without markers) and RC (curves
with markers) P-CPM schemes with $\mathcal{Y} \leq 64$.}
\label{fig:C64er-RECvsRC}
\end{figure}

In order to perform a fair comparison, the decoding complexity must be
kept into account.
Using definitions (\ref{eq:complexity-pcpm}) and (\ref{eq:complexity-sccpm}), the
decoding complexities of the two schemes have been computed.\
For the P-CPM scheme, the complexity of the MHOMS~\cite{bib:MHOMS} binary
decoder has been computed as follows:

\begin{equation}
\mathcal{Y}_{\rm SCCC} = 2 N_{it}\left(N_{so} + \frac{3}{2}
N_{si}\right)
\label{eq:SCCCComplexity}
\end{equation}

\noindent where the factor $3/2$ derives from the outer code rate of
the MHOMS scheme, which is 2/3.   Results are shown in
Tab.~\ref{tab:complexity}: we
observe that the complexity of the SC-CPM scheme is always larger than
that of the P-CPM scheme.   We define the complexity ratio $R_{\cal Y}
= \mathcal{Y}_{\rm SC-CPM} / \mathcal{Y}_{\rm P-CPM}$: its value is larger
than 2 and exceeds 3 for a 1 bits/s/Hz capacity with REC frequency pulse.
Higher coding gains could be achieved increasing the P-CPM complexity,
thus reducing the performance gap with respect to SC-CPM schemes.

\begin{table}[!ht]
\centering
\begin{tabular}{r|c||cc|c|c|c}
& \multirow{2}{*}{Capacity}& \multicolumn{3}{c|}{$P-CPM$} & $SC-CPM$ & Ratio\\
&  & $\mathcal{Y}_{SCCC}$ & $\mathcal{Y}_{CPM}$ &
$\mathcal{Y}_{P-CPM}$ & $\mathcal{Y}_{SC-CPM}$ & $R_{\cal Y}$\\
\hline \hline
\multirow{3}{*}{\begin{sideways}REC\end{sideways}} & 1   &  200  &  21  & 221 &  694 & 3.14  \\
& 1.5 &  200  &  41  & 241 &  709 & 2.94 \\
& 2   &  200  &  87  & 287 &  728 & 2.54 \\
\hline
\multirow{3}{*}{\begin{sideways}RC\end{sideways}} & 1   &  200  &  51  & 251 &  592 & 2.36  \\
& 1.5 &  200  &  34  & 234 &  565 & 2.41 \\
& 2   &  200  &  76  & 276 &  733 & 2.66
\end{tabular}
\caption{Complexity of P-CPM and SC-CPM schemes. ${\cal Y} = 64$.}
\label{tab:complexity}
\end{table}

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

The pragmatic approach to coded continuous-phase modulation (CPM) has been proposed
as a capacity-approaching low-complexity alternative to the serially-concatenated
CPM (SC-CPM) coding scheme.
After performing a selection of the best spectrally-efficient
CPM modulations to be embedded into SC-CPM schemes, the pragmatic
capacity  of CPM modulations has been evaluated and  optimized  through a careful
design of the mapping between input bits and CPM waveforms.
The so obtained schemes have been cascaded with an outer serially-concatenated convolutional
code to form a pragmatic coded-modulation system.
The resulting schemes have been shown to exhibit performance  close to the CPM capacity without
requiring iterations between the outer decoder and the CPM demodulator. As a result,
the receiver exhibits reduced complexity and increased flexibility due to the separation
of the demodulation and decoding functions.

\section*{Acknowledgements}

This work has been supported by Regione Piemonte under Contract E4.

\bibliographystyle{IEEEtran}

config: arxiv


TITLE: Consecutive numbers where their revers numbers are primes
QUESTION [0 upvotes]: Consider the consequtive numbers 1393 and 1394, when we revers their digits we get 3931 and 4931 which are  both prime numbers. Is this a well known property?
Here is my concerns:
Is there an infinite number of this numbers?
Let $k\geq 2$, for what values of $k$ there is $k$ consequtive numbers such that when we revers their digits we get prime numbers?
Any hint  or references would be helpful.

REPLY [1 votes]: Heuristically we should expect infinitely many for $k=2$.  If $x$ does not end in $0$ or $9$ (so its reverse and that of $x+1$ have the same number of digits) and $x \equiv 1 \mod 3$, the probability that the reverses of $x$ and of $x+1$ are both prime should be on the order of $1/(\log x)^2$, and
$\sum_x 1/(\log x)^2 = \infty$.  In fact these numbers should have roughly similar order of abundance to twin primes.
I found $1955$ examples below $10^5$ and $12943$ below $10^6$ (compare to $1224$ and $8169$ sets of twin primes in the same intervals).
I'm surprised the OEIS doesn't seem to have the sequence of $n$ such that the reversals of $n$ and $n+1$ are primes ($2,13,16,30,31,34,37,70,\ldots$).  Well, maybe it will soon...
EDIT: Now it does: A257636


TITLE: What is negation of the following sentence?
QUESTION [4 upvotes]: What is negation of All birds can fly.
The question seems bit funny but i don't know which of the following two sentences is correct:

Some birds can not fly
There is at least one bird which can not fly.

Both the sentence seems almost logically same. But which of the following is true.
In book correct option out of four is sentence 1.

REPLY [2 votes]: Translated in to set-language: "Every element of the set of birds $B$ belongs to the set of animals that fly $F$". So this translates as $B \subset F$. Negating this would be the equivalent of saying $B \nsubseteq F$. So in set language this would translate:"At least one bird cannot fly"(there is an element of $B$ that does not ly in $F$).


TITLE: Calculate power for a non-constant force
QUESTION [0 upvotes]: Suppose there is a body of mass $m$ in a horizontal plane, with initial velocity $v_0$, that is subjected to a force $F(t)$ (the force depends on time). How can I calculate the power developed by the force at any given time?

REPLY [0 votes]: P = W/T,
Instantaneous power is $$F.dx/dt
                    = Fv (dx/dt = v)$$
I think the the $F(t)$ will be known which can
give you the equation of v(t) by
$ v_0 + \int F/m .dt$
in $F.v$
Which is $$P =  F.(v_0+ \int a(t).dt  $$
$$a =  F/m$$
$$P = F.(v_0 + 1/m \int F.dt) $$


TITLE: How do you solve heavy equations in a organised and efficient way?
QUESTION [1 upvotes]: I've made countless small mistakes on my exams that would hurt my grade. I have not been able to find a good guide on how to structure equation solving on paper so I thought I'd ask here.
I've added a paper where I've solved for the possible max/min points in a multivariable equation. The paper is in my view very unorganized and I, therefore, wanted to ask you guys how you would solve it in a structural and organized way. Thanks in advance.

REPLY [0 votes]: My suggestion is to not forget that we are working with a system of two equations:
$$
\begin{cases}
f_x=0\\
f_y=0
\end{cases} \quad \iff \quad
\begin{cases}
y(2x+y^2+y)=0\\
x(x+3y^2+2y)=0
\end{cases}
$$
so:
1) from the first equation we have $y=0$ that, subtituting in the second equation gives $x=0$ and we have found one solution of the system : $(x_1,y_1)=(0,0)$
2) from the second equation we have $x=0$ ad substituting in the first we find $y=-1$ or $y=0$. So we have another, different, solution of the system : $(x_2,y_2)=(0,-1)$ (note that, from here we have also the first solution $(x_1,y_1)$ that we just known).
3) if $x\ne0$ and $y \ne 0$ the system becomes:
$$\begin{cases}
2x+y^2+y=0\\
x+3y^2+2y=0
\end{cases}
$$
that, substituting $x$ from the second equation in the first, gives a new solution: $(x_3,y_3)=(\frac{3}{25},-\frac{3}{5})$


TITLE: Multiplicative Inverses in Non-Commutative Rings
QUESTION [2 upvotes]: My abstract book defines inverses (units) as solutions to the equation $ax=1$ then stipulates in the definition that $xa=1=ax$, even in non-commutative rings. But I'm having trouble understanding why this would be true in the generic case.
Can someone help me understand?
Book is "Abstract Algebra : An Introduction - Third Edition" By Thomas W. Hungerford. ISBN-13: 978-1-111-56962-4 (Chapter 3.2)
Edit:
Ok, I think I got it. Left inverses are not necessarily also right inverses. However, if an element has a left inverse and a right inverse, then those inverses are equal:
$$ lx = 1 \\ xr = 1\\ lxr = r \\ lxr = l \\ r = l $$
Source:http://www.reddit.com/r/math/comments/1pdeyf/why_do_multiplicative_inverses_commute_even_in/cd17pcb
Also, I assumed that 'unit' was synonymous with 'has a multiplicative inverse'. It is not; a 'unit' is an element that has both a left and right inverse, not just an inverse in general.

REPLY [2 votes]: Note: In a ring R with 1, if EVERY non-zero element x has a left inverse, then R is a division ring (so ax=1 implies xa=1).
More generally, if EVERY non-zero element has either a left or a right inverse, then R is a division ring.


TITLE: Lattice Point Geometry
QUESTION [0 upvotes]: A student of mine asked me the following question:
Three lattice points (points with integer coordinates) are chosen at random with replacement (meaning you can select the same point more than once) in the interior of the square defined by $-99 \le x \le 100$ and also $-99 \le y \le 100$. The probability that the area of the triangle thus formed (which may be degenerate) is an integer can be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
I know the answer. However, I need an explanation with the answer.
Thank you!
Also I have just a few hours before I go back to teach him.

REPLY [0 votes]: Let the three lattice points be $(a_1, b_1)$, $(a_2, b_2)$, and $(a_3, b_3)$. By the shoelace formula (as @Jeannette said), the area of the triangle is
$\frac{|a_1b_2 + a_2b_3 + a_3b_1 - b_1a_2 - b_2a_3 - b_3a_1|}{2}.$
Note that whether or not the triangle has integer area depends only on the parity of the points.
From $-99 \leq x \leq 100$ and $-99 \leq y \leq 100$, we see that each coordinate has an equal probability of being even or odd. Hence, all three points are chosen from the four parity combinations $(even,even)$, $(even,odd)$, $(odd,even)$, and $(odd,odd)$, with replacement. It is true that the triangle have non-integer area iff they are all of different parity combinations (think about it).
We can choose any of the four parity combinations for the first point. The second point has a probability of $\frac{3}{4}$ of having a different parity from the first point. The third point has a probability of $\frac{1}{2}$ of having a different parity than the other two points. Hence, the probability that all three points have different parities (and therefore the triangle has non-integer area) is $\frac{3}{4} \cdot \frac{1}{2} = \frac{3}{8}$.
Therefore, the probability that the triangle has integer area is $1 - \frac{3}{8} = \frac{5}{8}$. The final answer is $5 + 8 = \boxed{13}$.


TITLE: Does writing $f(x)\sim \ell$ have a sense?
QUESTION [4 upvotes]: If $\lim_{x\to a}f(x)=\ell$, is it correct to say that $f(x)\sim_a \ell$ ? I would say yes since $\lim_{x\to a}\frac{f(x)}{f(a)}=1$, but on a test I wrote $e^{-t}\sim_0 1$ and the corrector said that it's wrong to say that.
So, what's wrong ?

REPLY [1 votes]: Yes I'd say it's true. From the property of the limit we get $|f(x)-L|<\varepsilon$, so there are two bounds, say $L-\frac{1}{n}, L+\frac{1}{n}$ such that $f(x)$ is within these bounds. Hence the ratio is 1 by squeeze lemma.


TITLE: Impact of constraints on convex optimization
QUESTION [3 upvotes]: Let me start by saying I know almost nothing about optimization so please bear with me. Basically, I am wondering whether it is possible to solve a problem with two constraints by solving the problem with each constraint individually and somehow combining the solutions. However, I'm having trouble visualizing the scenario. More specifically:
Suppose $V$ is some Banach space and $f:V \to \mathbb{R}$ is convex and continuous. Furthermore, let $K_1, K_2 \subset V$ be convex and consider the solutions $x_i$ of the optimization problems
$$\quad \min_{x \in K_i}  f(x).$$
For $K = K_1 \cap K_2$ let $x$ solve
$$\min_{x \in K} f(x).$$

Is $x$ the projection of $x_1$ onto $K_2$?
Is $x$ a linear combination of $x_1$ and $x_2$?
Are there any relations between $x$, $x_1$ and $x_2$ at all?
Does the situation change if we further restrict $f$? For example to be quadratic?

Thank you very much in advance.

REPLY [2 votes]: For general convex sets, the sorts of simple relationships you're suggesting don't exist. I find it helps to draw two dimensional pictures to see why. (Even if the space is dimension higher than 2, we could restrict the problems to the plane spanned by $x_1,x_2,x$ and none of the answers to your questions would change.) Here's an example of what can happen even in the simple case where $f$ is linear:

The red arrow is the direction of the negative gradient of $f$. Minimizing $f$ means moving as far as possible in that direction. To address your specific questions:

No. In this case, the projection of $x_2$ onto $K_1$ is not in $K_2$, so it can't equal $x$. The projection of $x_1$ onto $K_2$ is in $K$, but still doesn't equal $x$
The origin is not depicted in the diagram, but if $x_1$ and $x_2$ are collinear with the origin, then $x$ is not a linear combination of $x_1$, $x_2$.
The only easy, simple relationship is that $f(x_1) \le f(x)$ and $f(x_2) \le f(x)$.
In this example, the function is linear. That's about as simple as you can get without making the problem trivial.

That said, there are methods (like ADMM) that can use solvers for subproblems to converge to a solution of a larger problem.