Conjugate-Root Offset-QAM for Orthogonal Multicarrier Transmission
% \IEEEauthorblockN{Author A
Author B
Author C
\IEEEauthorblockN{Maximilian Matth\'{e
% \IEEEauthorblockN{Author A
Author B
Author C
\IEEEauthorblockN{Maximilian Matth\'{e
% Problem Current implementations of OFDM/OQAM are restricted to band-limited symmetric filters. % Solution To circumvent this, non-symmetric conjugate root (CR) filters are proposed for OQAM modulation. The system is applied to Generalized Frequency Division Multiplexing (GFDM) and a method for achieving transmit diversity with OQAM modulation is presented. % Results. The proposal reduces implementation complexity compared to existing works and provides a more regular phase space. GFDM/CR-OQAM outperforms conventional GFDM in terms of symbol error rate in fading multipath channels and provides a more localized spectrum compared to conventional OQAM. %Impact. \begin{comment This paper presents a novel approach to multicarrier Offset-QAM (OQAM) with conjugate root (CR) filters, which provides a more regular phase space. CR-OQAM is analyzed in the context of Generalized Frequency Division Multiplexing (GFDM), a waveform proposed for 5G applications and a simple method for combining GFDM/CR-OQAM with Alamouti's space-time coding is presented. The symbol error rate (SER) performance in fading multipath channels is evaluated, showing that GFDM/CR-OQAM outperforms conventional GFDM. \CR{update \end{comment
1.tex
% Section Introduction
%\CMax{I dont like this intro. It has been written in a rush}

With%
\let\thefootnote\relax\footnote{Manuscript received..., revised\\
  The responsible editor during processing at IEEE WCL was ...\\
  Here goes some more editorial text...\\
  The DOI of this document is ....\\
  This work has been performed in the framework of the FP7 project
  ICT-619555 RESCUE, which is partly funded by the European Union.\\
  M. Matth\'e and G. Fettweis are with Vodafone Chair Mobile
  Communication Systems, Technische Universität Dresden, Germany
  (e-mail: maximilian.matthe@ifn.et.tu-dresden.de;
  fettweis@ifn.et.tu-dresden.de)} 5G on the horizon, new waveforms for
the PHY layer are investigated that are suitable for the upcoming
requirements \cite{Wunder2014}.  In particular, good \ac{TFL} of the
transmit signal is required to cope with asynchronicities
\cite{Floch95} and to provide a low \ac{OOB} radiation, which is
needed for spectral agility and carrier aggregation.  Filtered
\ac{MC} systems \cite{Chang1966} provide the means for good \ac{TFL}
by adaptation of the prototype filter and spectral agility is achieved
by switching on and off certain subcarriers.  High spectral efficiency
is important to serve the increased demand for high speed data access
\cite{Wunder2014}.  Hence, a future waveform should transmit symbols
at the Nyquist rate in order to not waste valuable time-frequency
resources.

Even in a distortion-free channel, the \ac{BLT}
prohibits the distortion-free reconstruction of
complex valued-symbols sent at Nyquist rate when using filters with
good \ac{TFL} \cite{Benedetto1998}.
%
Hence, in terms of \ac{SER}, QAM \ac{MC} systems with good \ac{TFL} perform
worse than orthogonal systems without good \ac{TFL}.
%
One possibility to circumvent the \ac{BLT} is to transmit real-valued
symbols at twice the Nyquist rate, which was initially proposed
in~\cite{Chang1966}.  This gave rise to the well known
OFDM/OQAM~\cite{Siohan2002} \ac{MC} system where both
orthogonality and good \ac{TFL} is kept by using Offset-QAM (OQAM),
which is proposed for upcoming 5G networks \cite{Wunder2014}.

\MMa[cSym1]{In current OQAM proposals, the employed real-valued
  prototype filter is required to be bandlimited and symmetric and a
  phase shift of $\tfrac{\pi}{2}$ is maintained between adjacent
  subcarriers. In \cite{Vangelista2001}, low-complexity
  implementations of these schemes are described and also a method is
  proposed to relax the requirement of symmetry of the employed
  prototype filter which is advantageous since it is shown in
  \cite{Feher1993}, that non-symmetric filters can be more robust
  against multipath fading. However, the approach presented in
  \cite{Vangelista2001} suffers from an increased implementation
  complexity compared to conventional OQAM since it requires to use
  two \acp{PPN}. }

In this paper we contribute another version of \ac{MC} OQAM
modulation, named \ac{CR} Offset-QAM (CR-OQAM), which allows to use
the class of non-symmetric CR filters, removing the $\tfrac{\pi}{2}$
phase shift and providing means for an easier implementation compared
to \cite{Vangelista2001}.
%
We further apply CR-OQAM to \ac{GFDM} \cite{Michailow2014}, which is another 5G candidate
waveform that normally uses QAM modulation and hence suffers from the
\ac{BLT}. With CR-OQAM, we provide a well-localized orthogonal system,
improving the \ac{SER} performance in fading multipath channels
compared to standard \ac{GFDM}.  Additionally, time-reversal
space-time-coding (TR-STC) \cite{Matthe2015a} is applied to
GFDM/CR-OQAM to exploit multi-antenna diversity, showing that in GFDM
STC can be combined with OQAM modulation which is a known issue
in OFDM/OQAM \cite{Lele2010}.  \MMa[Siohan]{In \cite{Lin2014},
  FBMC/COQAM is proposed, being equivalent to GFDM using
  OQAM. However, compared to the present paper, neither CR-OQAM nor
  STC is covered therein.}

The remainder is organized as follows: Sec. II describes the
conventional OQAM transceiver, whereas the proposed modification is
presented in Sec. III. Sec. IV describes the application of CR-OQAM to
GFDM and performance simulations are provided in Sec. V. Finally,
Sec. VI concludes.


\begin{comment}
- 5G requirements
- good localization, spectral agility
  --> filtered multicarrier
- high SE
  --> critical sampling
- Balian-Low
  --> QAM difficult
  --> Use OQAM modulation
- OFDM/OQAM and FBMC/OQAM achieve full SE, use OQAM
- GFDM uses QAM, circumvents BLT for some configurations

- aim:
 - provide a new OQAM scheme with a more regular phase space
 - apply OQAM to GFDM to completely circumvent the BLT
    --> more flexibility regarding parameter configurations
 - Simple technique for STC with OQAM
\end{comment}


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2.tex
\section{Conventional OFDM/OQAM}
%\section{Conventional OQAM}

% - initially found by Chang \cite{Chang1966} and Saltzberg \cite{Saltzberg1967} with bandlimited filters
% - standard QAM and maximum spectral efficienly: bad localization (Balian-Low)
% - with OQAM possiblity for localization in time and frequency,
% - optimization of filters carried out (for strongest localization)
% - OFDM/OQAM \cite{Siohan2002}

% - double symbol rate (1/(2T)), but only real symbols transmitted. Subcarrier distance 1/T
% - phase shift of pi/2 between subcarriers
% - In Gabor Theory reasoned by Wilson Bases \cite{Bolcskei2003}

\def\oqamheight{4cm}
\begin{figure*}[t]
  \centering
  \resizebox{!}{3.6cm}{\includegraphics{figures/conventionalOqam}}
  \caption{Block diagram of the OFDM/OQAM ($\phi=1$) and OFDM/CR-OQAM
    ($\phi=0$) transceiver.}
  \label{fig:convOqam}
\end{figure*}

In OFDM/OQAM, complex-valued data symbols $c_{k,m}$ are transmitted on
K subcarriers, where real and imaginary part are offset by
$\tfrac{T}{2}$ where $T$ is the symbol duration.  Each symbol is
pulse-shaped with a symmetric, real-valued pulse shaping filter $g(t)$
\cite{Chang1966,Siohan2002}. The transmission equation is given by
%
\parbox[t]{\linewidth}{\small
\begin{align}
  x(t) &= \sum_{k=0\atop m\in\mathbb{Z}}^{K-1}(c_{k,m}^{\text{R}}g(t-mT)+jc_{k,m}^{\text{I}}g(t-mT-\tfrac{T}{2}))j^kw^{kt}
\end{align}
}
%
where $w=\exp(j2\pi F)$, $F=\tfrac{1}{T}$, and $c^{\text{R}}_{k,m}$ and
$c^{\text{I}}_{k,m}$ are real and imaginary part of $c_{k,m}$,
respectively. Note that, due to the factor $j^k$, adjacent subcarriers
differ by a phase shift of $\tfrac{\pi}{2}$.

At the receiver, matched filtering is carried out, i.e.
%
\begin{align}
  \hat{c}^{\text{R}}_{k,m} &= \Re (x(t)j^{-k}w^{-kt}*g(-t))|_{t=mT}\\
  \hat{c}^{\text{I}}_{k,m} &= \Im (x(t)j^{-k}w^{-kt}*g(-t))|_{t=(\oh+m)T},
\end{align}
%
where convolution $x(t)*y(t)$ is defined as
%
\begin{align}
  (x(t)*y(t))|_{t=\tau}&=\int_{-\infty}^{\infty}x(t)y(\tau-t)dt.
\end{align}
%
The corresponding block diagram of the OFDM/OQAM transceiver is
presented in Fig. \ref{fig:convOqam}. For perfect reconstruction, the
orthogonality conditions between the $k$th and $(k+\kappa)$th subcarrier
that are given by \cite[eq. (13)]{Siohan2002}
%
\begin{align}
\label{eq:o1} \Re& \bigl\{(g(t)j^{-\kappa}w^{-\kappa t}*g(-t))|_{t=mT\vphantom{\oh}}\bigr\}&&=\delta(\kappa,m)\\
\label{eq:o2} \Re& \bigl\{(jg(t)j^{-\kappa}w^{-\kappa t}*g(-t))|_{t=(\oh+m)T}\bigr\}&&=0\\
\label{eq:o3} \Im& \bigl\{(g(t)j^{-\kappa}w^{-\kappa t}*g(-t))|_{t=(\oh+m)T}\bigr\}&&=0\\
\label{eq:o4} \Im& \bigl\{(jg(t)j^{-\kappa }w^{-\kappa
    t}*g(-t))|_{t=mT\vphantom{\oh}}\bigr\}&&=\delta(\kappa ,m)
\end{align}
%
must be fulfilled for all $\kappa,m$.  With the relations
\mbox{$\Re\{ja\}=-\Im\{a\}$} and $\Im\{ja\}=\Re\{a\}$, (\ref{eq:o1})
and (\ref{eq:o2}) are equivalent to (\ref{eq:o4}) and (\ref{eq:o3}),
respectively.  These conditions are in particular fulfilled, when
$g(t)$ is a symmetric, half-Nyquist filter with band-limitation $G(f)=0,
|f|\geq{}F$ \cite{Floch95}.

\noindent The convolution is expressed with the Fourier transform
$\mathcal{F}$ by
%
\begin{align}
  (g(t)w^{-kt}*g(-t))|_{t=\tau}&=\mathcal{F}^{-1}\{S_k(f)\}(\tau)=s_k(\tau),
\end{align}
%
where $S_k(f)=G(f-kF)G^*(f)$ is the spectrum of the intercarrier
interference (ICI) from the $k'$th to the $(k'+k)$th subcarrier,
$s_k(t)$ is the corresponding time domain function, $G(f)$ is the
Fourier transform of $g(t)$ and $(\cdot)^*$ denotes complex
conjugation.  Fig. \ref{fig:adj} depicts $s_1(t)$ for a
\ac{RRC} filter with roll-off $\alpha=0.75$.

% In \cite{Floch95} it is
% stated that the orthogonality conditions are fulfilled for any even and
% bandlimited function that is a half-Nyquist filter.

% When $G(f)$ is a bandlimited filter according to $G(f)=0, |f|>F$, the
% orthogonality conditions are clearly fulfilled for $|k|\geq2$. For
% $k=0$, (\ref{eq:o1}) is fulfilled if $|G(f)|^2$ is a Nyquist filter
% [ref!] and (\ref{eq:o3}) is fulfilled, if $g(t)$ is a real-valued
% function. The case $|k|=1$ is depicted in
% Fig. \ref{fig:interference}(a) for $g(t)$ being a root raised cosine
% filter with rolloff $\alpha=0.75$.  Note that $S_1(f)$ is purely
% real. Looking at its time-domain representation $s_1(t)$, the
% orthogonality conditions (\ref{eq:o2}) and (\ref{eq:o4}) are clearly
% fulfilled.

% Hence, the OQAM becomes orthogonal when using the
% bandlimited root raised cosine filter as the pulse shaping filter.

\begin{figure*}[t]
  \centering
  \resizebox{!}{4.3cm}
  {
   \subfloat[\ac{ICI} of \ac{RRC}.]{\label{fig:adj}
   \includegraphics[width=0.33\linewidth]{figures/adjacentChannel}}
   \subfloat[\ac{RRC} and \ac{CRRC} impulse and frequency response.]{\label{fig:tdfd}
   \includegraphics[width=0.33\linewidth]{figures/tdfd}}
   \subfloat[\ac{ICI} of \ac{CRRC}.]{\label{fig:adjcr}
   \includegraphics[width=0.33\linewidth]{figures/adjacentChannelCR}}}
 \caption{Solid and dashed lines represent real and imaginary part,
   respectively. (a), (c) ICI of adjacent channel. (b) Filter response of RRC and CRRC
   filters.}
  \label{fig:interference}
\end{figure*}

% \begin{figure*}[t]
%   \centering
%   \resizebox{!}{\oqamheight}{\includegraphics[height=4cm]{figures/modifiedOqam}}
%   \caption{Block diagram of Conjugate-Root OQAM transceiver.}
%   \label{fig:CRoqam}
% \end{figure*}


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3.tex
\section{Conjugate-Root Multicarrier OQAM}
In literature, mostly symmetric half-Nyquist filters are employed for
\ac{MC} OQAM systems. We propose the application of
non-symmetric \ac{CR} filters \cite{Demeechai1998,Tan2004}
for \ac{MC} OQAM systems.

% Demeechai \cite{Demeechai1998} formalized the concept of conjugate
% root (CR) filters.

% Xia proposed a family of non-symmetric pulse shaping filters that
% fulfill the 1st Nyquist criterion [ref!] with both matched filtering and
% unmatched lowpass filtering \cite{Xia1997}. Demeechai
% \cite{Demeechai1998} generalized these filters to the concept of what
% are now known as conjugate root (CR) filters \cite{Tan2004}.


Assume an even $H(f)$ with band limitation $H(f)=0$ for
$|f|\geq F$. Then, $H(f)$ fulfills the 1st Nyquist criterion
\cite{Demeechai1998} iff
%
\begin{align}
  \label{eq:NQ}
  \forall  f\in[0,F]: H(f)+H(f-F)=1.
\end{align}
%
Its corresponding half-Nyquist filter is given by
\mbox{$G(f)=\sqrt{H(f)}$} where the most prominent example is the
symmetric \ac{RC} and \ac{RRC} pair. The
according non-symmetric CR filter $G\C(f)$ is constructed by
%
\begin{align}
  G\C(f) &=
  \begin{cases}
    H(f) + j \sqrt{(1-H(f))H(f)} & f\geq 0\\
    H(f) - j \sqrt{(1-H(f))H(f)} & f< 0
  \end{cases}
  \label{eq:CR}
\end{align}
%
with impulse response $g\C(t)=\mathcal{F}^{-1}\{G\C(f)\}$.  Note that
both $G\C(f)$ and $G\C(f)(G\C(f))^*$ are Nyquist filters
(cf. (\ref{eq:NQ})).

Fig.~\ref{fig:tdfd} shows the \ac{RRC} and \ac{CRRC} filter
response.  The \ac{ICI} $S_1\C(f)$ between adjacent subcarriers with \ac{CR}
filters equals
%
\begin{align}
  S_1(f) %&=G(f-F)G^*(f)\\
  &=\sqrt{H(f)H(f-F)}\\
  % \begin{cases}
  %   \sqrt{H(f)H(f-F)} & 0\leq f\leq F\\
  %   0 & \text{else}
  % \end{cases}\\
  S_1\C(f)&=G\C(f-F)[G\C(f)]^*\\ \label{eq:jS1}
  &= jS_1(f),
\end{align}
%
where (\ref{eq:jS1}) follows from (\ref{eq:CR}) and (\ref{eq:NQ}). Therefore, since
%
\begin{align}
  s_1\C(t)=js_1(t),
  \label{eq:icitd}
\end{align}
%
the real and imaginary part of the \ac{ICI} are exchanged when using a \ac{CR}
filter compared to its standard half-Nyquist version.  Both $S\C_1(f)$ and
$s\C_1(t)$ are presented in Fig. \ref{fig:adjcr}.

%\subsection{OQAM with conjugate root filters}
Hence, if $g(t)$ fulfills (\ref{eq:o1})--(\ref{eq:o4}), the according $g\C(t)$ fulfills
% From (\ref{eq:icitd}) and Fig. \ref{fig:interference}(b), the following
% orthogonality conditions can be derived:
%
\begin{align}
\label{eq:co1} \Re& \bigl\{(g\C(t)w^{-kt}*g\C(-t))|_{t=mT\vphantom{\oh}}\bigr\}&&=\delta(k,m)\\
\label{eq:co2} \Re& \bigl\{(jg\C(t)w^{-kt}*g\C(-t))|_{t=(\oh+m)T}\bigr\}&&=0.%\\
% \label{eq:co3} \Im& \left\{(g\C(t)w^{-kt}*g\C(-t))|_{t=(\oh+m)T}\right\}&&=0\\
% \label{eq:co4} \Im& \left\{(jg\C(t)w^{-kt}*g\C(-t))|_{t=mT\vphantom{\oh}}\right\}&&=\delta(k,m).
\end{align}
%
Note, that compared to (\ref{eq:o1})--(\ref{eq:o4}) the factor $j^k$
has been removed.
% Again, (\ref{eq:co1}) and (\ref{eq:co2}) are
% equivalent to (\ref{eq:co4}) and (\ref{eq:co3}), respectively and are
% clearly fulfilled for $k\geq 2$ because $G^C(f)$ is bandlimited. For
% $k=0$, (\ref{eq:co1}) is fulfilled since $H(f)$ is a Nyquist filter
% and (\ref{eq:co2}) is fulfilled since $g^C(t)$ is real-valued. For
% $k=1$, (\ref{eq:co1}) and (\ref{eq:co3}) follow from (\ref{eq:o1}),
% (\ref{eq:o3}) and (\ref{eq:icitd}).
Consequently, an OFDM/CR-OQAM system that uses a
\ac{CR} filter $g\C(t)$ can be described by the modulation equation
%
\parbox[t]{\linewidth}{\footnotesize
\begin{align}
  x(t) &= \sum_{k=0\atop m\in\mathbb{Z}}^{K-1}(c_{k,m}^{\text{R}}g\C(t-mT)+jc_{k,m}^{\text{I}}g\C(t-mT-\tfrac{T}{2}))w^{kt},
\end{align}
}
%
the demodulation equations
%
\begin{align}
  \hat{c}^{\text{R}}_{k,m} &= \Re (x(t)w^{-kt}*g\C(-t))|_{t=mT}\\
  \hat{c}^{\text{I}}_{k,m} &= \Im (x(t)w^{-kt}*g\C(-t))|_{t=(\oh+m)T}
\end{align}
%
and the block diagram in Fig. \ref{fig:convOqam} with $\phi=0$.
% Note that, compared to Fig. \ref{fig:convOqam}, the multiplication
% with $j^{\pm k}$ is not required.
% \CMax{State that any oqam system with
% bandlimited G(f) can be converted to a modified OQAM system using
% $g^c(t)$ as the filter.}

% \begin{figure}[t]
%   \centering
%   \includegraphics[height=3.3cm]{figures/phasespace}
%   \caption{Time-frequency phase-space for OQAM and CR-OQAM.}
%   \label{fig:phasespace}
% \end{figure}

\begin{figure*}[t]
  \centering
  \subfloat[Time-frequency phase-space.]{\label{fig:phasespace}\includegraphics[height=3.8cm]{figures/phasespace}}
  \subfloat[SER performance in fading multipath channels, 16-QAM.]{\label{fig:results}\includegraphics[height=3.8cm]{figures/simulation}}
  \subfloat[Comparison of PSD.]{\label{fig:spectrum}\includegraphics[height=3.8cm]{figures/spectrum}}
   \caption{Time-frequency phase space and performance comparison of OQAM and CR-OQAM.}
\end{figure*}

\tikzexternaldisable The time-frequency phase space for OFDM/OQAM and
OFDM/CR-OQAM is shown in Fig. \ref{fig:phasespace} where real and
imaginary values are depicted with \scalebox{0.7}{\R}~and \scalebox{0.7}{\I}, respectively. Due to
the missing $\tfrac{\pi}{2}$ phase shift between subcarriers, the grid
is more regular for the CR-OQAM case. % The
% required phase shift of $\tfrac{\pi}{2}$ between the subcarriers for
% conventional OQAM is nicely seen, whereas it is not present in the
% CR-OQAM case and the CR-OQAM becomes more regular.
Hence, OFDM/CR-OQAM can be seen as two parallel OFDM/PAM systems
transmitting with a time offset of $\tfrac{T}{2}$ and a
phase shift of $\tfrac{\pi}{2}$.  \tikzexternalenable \MMa[phasespace]{The
  more regular phase space can lead to simpler equalization and pilot
  design and theoretic analysis might be easier to accomplish.}% This understanding can
% simplify the implementation of CR-OQAM compared to conventional
% OQAM.

  \MMa[Laurenti1]{ In \cite{Vangelista2001} conjugate OFDM/OQAM is
    proposed, where forward-time and time-reversed transmit and
    receive filters alternate along the subcarriers, which is clearly
    different compared to the present proposal.  Using this
    approach, the symmetry requirement of the prototype filters is
    relaxed, but still a phase shift between subcarriers is kept,
    hence keeping the phase space equal to conventional OQAM.}\MMa[Laurenti2]{Also in
    \cite{Vangelista2001}, low-complexity formulations for
    conventional and conjugate OQAM are given.  It is stated that due
    to the use of alternating filters for conjugate OQAM, the system
    requires two \acp{PPN} for implementation, which increases the
    complexity.  In the present approach, the symmetry requirement is
    relaxed by using \ac{CR} filters and changing the transmit phase
    space.  Since only one prototype filter type is used, only one
    \ac{PPN} is required, leading to the same implementation
    complexity as conventional OQAM in \cite{Vangelista2001}.  However,
    in \cite{Vangelista2001} the use of one N/2-point complex DFT per
    symbol is required, which would be changed to one N-point real DFT
    per symbol due to the changed phase space for OFDM/CR-OQAM
    implementations.}



\begin{comment}
  - block diagram of 2x PAM
  - remove GFDM boilerplate, reference TCOM
  - remove TR-STC boilerplate, reference TWC
  - provide list of advantages when using OQAM
  - add figure/graph for non-STC GFDM.
  - should be done then
\end{comment}


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4.tex
% Section GFDM and OQAM
\section{Application of CR-OQAM in GFDM}
OFDM/OQAM is a streaming based, filtered \ac{MC} system, where every
symbol overlaps with its adjacent symbols in time.  \ac{GFDM}
\cite{Michailow2014} is a filtered \ac{MC} system, where circular
convolution is applied instead of linear.  Hence, its transmit signal
exhibits a block structure and subsequent blocks can be decoupled by a
\ac{CP} to ease equalization.

\ac{GFDM} is modeled in discrete base band with sampling period
$T_s$. The transmit signal $\vec{x}$ is given by
%
\begin{align}
  \vec{x}&=\mat{A}\vec{d},
\end{align}
%
where the columns of the matrix $\mat{A}$ contain circular
time-frequency shifted versions of a prototype transmit filter $g[n]$
with distance $KT_s$ in time and $1/(KT_s)$ in frequency, where $K$ is
the number of subcarriers. $\vec{d}$ contains the complex-valued data
symbols to be transmitted with the block.  By appending a \ac{CP},
frequency domain channel equalization can be carried out at the
receiver, yielding the signal $\hat{\vec{x}}$ and \ac{ZF} or
\ac{MF} detection is applied
%
\begin{align}
  \hat{\vec{d}}_{\text{ZF}} &= \mat{A}^{-1}\hat{\vec{x}} & \hat{\vec{d}}_{\text{MF}} &= \mat{A}^{\text{H}}\hat{\vec{x}},
\end{align}
%
where $(\cdot)^\text{H}$ denotes Hermitian conjugate.  A main property
of \ac{GFDM} is its good \ac{TFL} of the transmit filter, which allows to
achieve a low \ac{OOB} radiation and robustness against
asynchronicity \cite{Michailow2014}. However, when using QAM
modulation, the \ac{BLT} prohibits orthogonality completely,
which impacts \ac{MF} performance while \ac{ZF} detectors introduce
noise-enhancement and only exist for certain parameter configurations
\cite{Matthe2014}.  Hence, with perfect synchronization, SER
performance of \ac{GFDM} is worse compared to an orthogonal system.

To circumvent this problem, OQAM modulation can be applied, which
provides orthogonality but still keeps the advantageous
property of good time-frequency localization. Looking at the phase
space of CR-OQAM (Fig. \ref{fig:phasespace}), the GFDM/CR-OQAM
modulator is given by
%
\begin{align}
  \vec{x}&=\mat{A}\Re\{\vec{d}\}+j \mathcal{C}_{\tfrac{K}{2}}(\mat{A}\Im\{\vec{d}\}),
\end{align}
%
where $\mathcal{C}_{u}(\cdot)$ denotes a circular rotation of its argument
by $u$ elements. At the receiver, the CR-OQAM detection with the
matched filter is simply achieved by
%
\begin{align}
  \Re\{\hat{\vec{d}}\}&=\Re\{\mat{A}^H\vec{x}\},&
  \Im\{\hat{\vec{d}}\}&=\Im\{\mat{A}^H\mathcal{C}_{-\tfrac{K}{2}}(\vec{x})\}.
\end{align}
%
Time-reversal space-time coding (TR-STC) \cite{Al-Dhahir2001} can be
applied to GFDM/CR-OQAM to provide transmit diversity in a fading
multipath environment. TR-STC was initially developed for
single-carrier systems, where two subsequent time domain codewords are
decoupled by a \ac{CP} and jointly space-time encoded.  For \ac{GFDM},
one block is treated as a single-carrier codeword and the
TR-STC is applied onto two subsequent \ac{GFDM} blocks \cite{Matthe2015a}. At the
receiver, both \ac{GFDM} blocks are recovered by
the TR-STC and processed by the \ac{GFDM} detector.  The approach
can be applied to both \ac{GFDM} and GFDM/(CR-)OQAM, but GFDM/(CR-)OQAM is
expected to outperform conventional \ac{GFDM} due to the kept
orthogonality.




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5.tex
% Section Simulation results
The \ac{SER} performance of \ac{GFDM} and GFDM/(CR-)OQAM with and
without TR-STC has been evaluated in Rayleigh fading multipath
channels.  The power delay profile of the channel between the transmit
and receive antennas is modelled with length 16 where the tap powers
decay linearly in log scale from 0dB to -16dB.  The \ac{GFDM}
parameters are given in Tab.  \ref{tab:gfdmconfig} and the simulation
results are presented in Fig. \ref{fig:results}, where perfect channel
state information was available at the receiver.
% For comparison, curves for
% TR-STC-GFDM with QAM and OQAM modulation are provided and also
% compared with theoretic equations.
\MMa[theory]{
The theoretic equations for GFDM/QAM are taken
from \cite{Matthe2015a}.  Note that as GFDM/(CR-)OQAM is orthogonal
it performs equal to OFDM/OQAM and corresponding equations are also
taken from \cite[eq. (19)]{Matthe2015a} with noise-enhancement $\xi=1$.}

GFDM/OQAM outperforms conventional \ac{GFDM} with and without TR-STC due to
the orthogonality introduced by the OQAM modulation. The performance gap equals the
noise-enhancement of \ac{GFDM}, which is $0.8$ dB in the presented
configuration.  Both \ac{GFDM} and GFDM/(CR-)OQAM show a diversity gain of 2
when combined with TR-STC.

\begin{table}[t]
  \centering
  \caption{GFDM simulation parameters}
  \begin{tabular}[t]{llccc}\toprule
   Parameter       & Symbol    & QAM  & OQAM & CR-OQAM \\ \midrule
   Subsymbol spacing  & $K$    & 64   & 64   & 64 \\
   Subsymbol count    & $M$    & 7    &  7   &  7 \\
   Filter          &  $g[n]$   & RC   & RRC  & CRRC \\
   Filter rolloff  & $\alpha$  & 0.5  &  1 & 1 \\
   Detector        &           & ZF   & MF   & MF  \\
%   Modulation      &           & 16-QAM & 16-QAM & 16-QAM\\
   \bottomrule
  \end{tabular}
  \label{tab:gfdmconfig}
\end{table}

% \begin{figure}[t]
%   \centering
%   \includegraphics[width=\linewidth]{figures/simulation}

%   \caption{Performance simulation in fading multipath channels.}
%   \label{fig:results}
% \end{figure}

\MMa[nonsymmPerf]{Since in the present configuration, OQAM and CR-OQAM
  both use the same equalization scheme, the benefits of
  non-symmetric filters proposed in \cite{Feher1993} cannot be shown
  in the \ac{SER} curve.}
\MMa[spectrum]{ Fig. \ref{fig:spectrum} shows the comparison of the
  \ac{PSD} of GDFM/OQAM and GFDM/CR-OQAM using one \ac{GS}
  \cite{Michailow2014}, where the application of \ac{CRRC}
  filters significantly reduces \ac{OOB} emission. As shown in
  \cite{Michailow2014}, high OOB emission is mainly due to abrupt
  signal changes at the boundaries of the GFDM block. By using
  \acp{GS} with QAM modulation and ISI-free filters,
  these discontinuities are avoided. However, due to the
  half-symbol shift when using OQAM with RRC filters, employing
  \acp{GS} does not remove signal discontinuities. On the other hand,
  as CRRC filters have zeros at twice the symbol rate, application of
  \ac{GS} efficiently reduces OOB emission, making the use of CR-OQAM
  advantageous over that of conventional OQAM.}

% \begin{figure}
%   \centering
%   \includegraphics[width=\linewidth]{figures/spectrum}
%   \caption{Comparison of PSD of GFDM with OQAM and CR-OQAM.}
%   \label{fig:spectrum}
% \end{figure}




%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End:

6.tex
% Conclusion
\MMa[conclusion]{
This paper has presented an alternative approach for the
implementation of Offset-QAM when using non-symmetric conjugate root
filters. With the proposal, no phase shift between
subcarriers is required, making the CR-OQAM time-frequency phase
space more regular and compared to existing work, implementation
complexity is reduced.
Orthogonality conditions have been stated for the OFDM/CR-OQAM system
and it was proven that \ac{CR} filters fulfill these.%\\
CR-OQAM has been applied to GFDM to create an orthogonal
system with good \ac{TFL}. Space-time code aiming for transmit diversity
was applied to GFDM/(CR-)OQAM and, outperforming the
conventional space-time coded GFDM system.  \ac{GS} were
efficiently employed to reduce \ac{OOB} emission when using CR
filters, which is not possible with conventional OQAM.}


%%% Local Variables:
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%%% TeX-master: "main"
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acronyms.tex
\begin{acronym}
  \acro{TFL}{time-frequency localization}
  \acro{BLT}{Balian-Low theorem}
  \acro{PPN}{poly-phase network}
  \acro{RRC}{root raised cosine}
  \acro{RC}{raised cosine}
  \acro{CR}{conjugate-root}
  \acro{CRRC}{conjugate RRC}
  \acro{CP}{cyclic prefix}
  \acro{ZF}{zero-forcing}
  \acro{MF}{matched filter}
  \acro{OOB}{out-of-band}
  \acro{SER}{symbol error rate}
  \acro{GFDM}{Generalized frequency division multiplexing}
  \acro{DFT}{discrete Fourier transform}
  \acro{FFT}{fast Fourier transform}
  \acro{ICI}{inter-carrier interference}
  \acro{PSD}{power spectral density}
  \acro{GS}{guard symbol}
  \acro{MC}{multicarrier}
\end{acronym}

tikzElements.tex
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\tikzstyle{shorten<>}=[shorten >=#1, shorten <=#1]

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\gdef\I{\tikz \filldraw (0,0) circle (4pt); }

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main.tex
\documentclass{IEEEtran}
\pdfoutput=1
%\documentclass[draftcls,12pt,onecolumn]{IEEEtran}
\input{preamble.tex}

\def\changesdir{changes/}
\newif\ifbypass
\bypasstrue  %% Uncomment to bypass comment highlighting and saving
\input{review.tex}

\makeatletter
\def\maketag@@@#1{\hbox{\m@th\normalfont\normalsize#1}}
\makeatother

\usetikzlibrary{external}
\tikzexternalize[prefix=tikztemp/]

\newcommand{\CMax}[1]{{\color{red}{[#1]}}}

\input{figures/tikzElements.tex}

\usepackage{standalone}

\title{Conjugate-Root Offset-QAM for Orthogonal Multicarrier Transmission}
% \author{
% 	\IEEEauthorblockN{Author A, Author B, Author C}\\
% 	\IEEEauthorblockA{Institution}\\
% 	\IEEEauthorblockA{EMAILS}
% }

\author{
	\IEEEauthorblockN{Maximilian Matth\'{e}, Gerhard Fettweis}\\
	% \IEEEauthorblockA{Vodafone Chair Mobile Communication Systems, Technische Universit\"{a}t Dresden, Germany}
	% \IEEEauthorblockA{\small\texttt{\{maximilian.matthe,fettweis\}\texttt{@ifn.et.tu-dresden.de}}}
}


\begin{document}
\input{acronyms.tex}

\maketitle


\begin{abstract}
%  \emph{Problem}
  Current implementations of OFDM/OQAM are restricted to band-limited
  symmetric filters.
% \emph{Solution}
  To circumvent this, non-symmetric
  conjugate root (CR) filters are proposed for OQAM modulation.  The system is applied
 to Generalized Frequency Division Multiplexing (GFDM) and a method
 for achieving transmit diversity with OQAM modulation is presented.
% \emph{Results}.
 The proposal reduces implementation complexity compared to existing
 works and provides a more regular phase space.
 GFDM/CR-OQAM outperforms conventional GFDM in terms of symbol
 error rate in fading multipath channels and provides a more localized
 spectrum compared to conventional OQAM.
%\emph{Impact}.
\begin{comment}
  This paper presents a novel approach to multicarrier Offset-QAM
  (OQAM) with conjugate root (CR) filters, which provides a more
  regular phase space.  CR-OQAM is analyzed in the context of
  Generalized Frequency Division Multiplexing (GFDM), a waveform
  proposed for 5G applications and a simple method for combining
  GFDM/CR-OQAM with Alamouti's space-time coding is presented.  The
  symbol error rate (SER) performance in fading multipath channels is
  evaluated, showing that GFDM/CR-OQAM outperforms conventional
  GFDM. \CR{update}
\end{comment}
\end{abstract}
\acused{GFDM}

\begin{IEEEkeywords}
  Quadrature Amplitude Modulation, Multicarrier Modulation, Space-Time Coding
\end{IEEEkeywords}

\section{Introduction}
% \begin{itemize}
% \item 5G-blabla
% \item OQAM long known (Chang/Saltzberg), FBMC most promiment candidate
% \item GFDM critical density, Balian-Low -> Singularity.
% \item with OQAM no singularity, no Balian-low
% \item non-symmetric easier to implement (more regular phase changes)
% \end{itemize}
\input{1.tex}

%\textbf{Description and Chain of argumentation}
% - Overall block diagram of conventional OQAM, present modulation equation\\
% - Conditions for orthogonality from k'th subcarrier to k'+k th subcarrier\\
% - Formulate these with IFFT(FFT*FFT).\\
% - Graphic showing that this works for RRC filter. (both in FD and
% TD). State that only adjacent subcarriers are examined, because of
% bandlimited RRC\\
% - Derive that the conjugate root-filters have the same interference
% shape, but multiplied by j, i.e. real and imaginary part are exchanged\\
% - Show graphic with interference from adjacent channel with Xia-filter\\
% - state that real and imaginary interference are exchanged\\
% - derive a following block diagram and modulation equations for the
% conjugate-root filter\\
%   - the $j^k$ factor is not necessary now\\

% - interpret the Xia OQAM as two independent PAM modulated filter
% banks, offset by half a symbol duration and one using real, one using
% imaginary data.\\


% \begin{itemize}
% \item Equations for modulation
% \item Real orthogonality
% \end{itemize}

\input{2.tex}

% \begin{itemize}
% \item Modulation Equation
% \item Picture showing real/imag-interference from adjacent subcarriers
%   of RRC and Xia
% \item Mathematical Proof that it works
% \item Picture with 1j1j1j1j for both rrc and xia
% \end{itemize}
\input{3.tex}

% \begin{itemize}
% \item short description of GFDM, focus on circularity
% \item recall problems with singularity
% \item modulation equations with A-matrix for Xia-filter
% \item equation for TR-STC
% \end{itemize}
\input{4.tex}

\section{Performance evaluation}
% \begin{itemize}
% \item Comparing standard GFDM (small NEF) with OQAM-GFDM
% \item AWGN, FSC
% \item TR-STC in fading multipath, compared to standard GFDM
% \end{itemize}
\input{5.tex}

\section{Conclusion}
% \begin{itemize}
% \item Non-symmetric filters work for OQAM, eases the distribution of
%   real and imaginary parts
% \item OQAM can increase GFDM performance, overcome Balian-Low
% \item OQAM works with TR-STC
% \end{itemize}
\input{6.tex}


\bibliographystyle{IEEEtran}
\bibliography{library}


\end{document}

preamble.tex
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_region_.tex
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%\documentclass[draftcls,12pt,onecolumn]{IEEEtran}
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\input{review.tex}

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\tikzexternalize[prefix=tikztemp/]

\newcommand{\CMax}[1]{{\color{red}{[#1]}}}

\input{figures/tikzElements.tex}

\usepackage{standalone}

\title{Conjugate-Root Offset-QAM for Orthogonal Multicarrier Transmission}
% \author{
% 	\IEEEauthorblockN{Author A, Author B, Author C}\\
% 	\IEEEauthorblockA{Institution}\\
% 	\IEEEauthorblockA{EMAILS}
% }

\author{
	\IEEEauthorblockN{Maximilian Matth\'{e}, Gerhard Fettweis}\\
	% \IEEEauthorblockA{Vodafone Chair Mobile Communication Systems, Technische Universit\"{a}t Dresden, Germany}
	% \IEEEauthorblockA{\small\texttt{\{maximilian.matthe,fettweis\}\texttt{@ifn.et.tu-dresden.de}}}
}


\begin{document}

\message{ !name(4.tex) !offset(-36) }
% Section GFDM and OQAM
\section{Application of CR-OQAM in GFDM}
OFDM/OQAM is a streaming based, filtered multicarrier system, where every
symbol overlaps with its adjacent symbols in time.  \ac{GFDM}
\cite{Michailow2014} is a filtered multicarrier system, where circular
convolution is applied instead of linear.  Hence, its transmit signal
exhibits a block structure and subsequent blocks can be decoupled by a
\ac{CP} to ease equalization.

\ac{GFDM} is modeled in discrete base band with a sampling period
$T_s$. The transmit signal $\vec{x}$ can be described by the matrix equation
%
\begin{align}
  \vec{x}&=\mat{A}\vec{d},
\end{align}
%
where the columns of the matrix $\mat{A}$ contain circular
time-frequency shifted versions of a prototype transmit filter $g[n]$
with distance $KT_s$ in time and $1/(KT_s)$ in frequency, where $K$ is
the number of subcarriers. $\vec{d}$ contains the complex-valued data
symbols to be transmitted with the block.  By appending a CP,
frequency domain channel equalization can be carried out at the
receiver, yielding the signal $\hat{\vec{x}}$ and zero-forcing (ZF) or
matched filter detection (MF) is applied
%
\begin{align}
  \hat{\vec{d}}_{\text{ZF}} &= \mat{A}^{-1}\hat{\vec{x}} & \hat{\vec{d}}_{\text{MF}} &= \mat{A}^{\text{H}}\hat{\vec{x}},
\end{align}
%
where $(\cdot)^\text{H}$ denotes Hermitian conjugate.  A main property
of GFDM is its good TFL of the transmit filter, which allows to
achieve a low out-of-band radiation and robustness against
asynchronicity \cite{Michailow2014}. However, when using QAM
modulation, the Balian-Low theorem prohibits orthogonality completely,
which impacts MF performance while ZF detectors introduce
noise-enhancement and only exist for certain parameter configurations
\cite{Matthe2014}.  Hence, with perfect synchronization, SER
performance of GFDM is worse compared to an orthogonal system.

To circumvent this problem, OQAM modulation can be applied, which
provides orthogonality but still keeps the advantageous
property of good time-frequency localization. Looking at the phase
space of CR-OQAM (Fig. \ref{fig:phasespace}), the GFDM/CR-OQAM
modulator is given by
%
\begin{align}
  \vec{x}&=\mat{A}\Re\{\vec{d}\}+j \mathcal{C}_{\tfrac{K}{2}}(\mat{A}\Im\{\vec{d}\}),
\end{align}
%
where $\mathcal{C}_{u}(\cdot)$ denotes a circular rotation of its argument
by $u$ elements. At the receiver, the CR-OQAM detection with the
matched filter is simply achieved by
%
\begin{align}
  \Re\{\hat{\vec{d}}\}&=\Re\{\mat{A}^H\vec{x}\}\\
  \Im\{\hat{\vec{d}}\}&=\Im\{\mat{A}^H\mathcal{C}_{-\tfrac{K}{2}}(\vec{x})\}.
\end{align}
%
% The filter $g[n]$ can be non-orthogonal and hence introduce
% self-interference when using the MF detector.  The ZF detector cancels
% this self-interference, but introduces noise-enhancement which
% degrades the performance compared to an orthogonal system.

% GFDM is treated in much detail in \cite{Michailow2014} and only a
% rough summary is provided here. The block based multicarrier system
% transmits $M$ subsymbols on $K$ subcarriers within one block with
% length $MK$ samples. Each block is extended by a cyclic prefix (CP) to
% combat multipath effects. The GFDM transmission equation of one block
% is given by

% \begin{align}
%   \label{eq:gfdmlin}
%   x[n] &= \sum_{k=0}^{K-1}\left(\sum_{m=0}^{M-1} d_{k}[m]\delta[n-mK]\circledast g[n]\right)e^{j2\pi\tfrac{kn}{K}}
% \end{align}

% where $n=0,\dots,MK-1$ is the sample index, $d_{k}[m]$ is the
% QAM-modulated data that is to be transmitted at the $k$th subcarrier
% in the $m$th subsymbol, $g[n]$ is the transmitter prototype filter and
% $\circledast$ denotes circular convolution with respect to the block
% length $KM$. (\ref{eq:gfdmlin}) can be reformulated into a matrix
% equation by

% \begin{align}
%   \vec{x}&=\mat{A}\vec{d},
% \end{align}

% where $\mat{A}$ contains all time-frequency shifts of $g[n]$ and
% $\vec{d}$ contains the complex-valued data \cite{Michailow2014}. After
% a CP is added to the signal, it passes a fading multipath channel. At
% the receiver side, the CP is removed and frequency domain equalization
% is carried out to yield the signal $\hat{\vec{x}}$. Afterwards,
% different receiver operations can be carried out \cite{Michailow2014},
% where the matched filter (MFR) and zero-forcing (ZFR) receiver are given
% by

% \begin{align}
%   \hat{\vec{d}}_{MFR} &= \mat{A}^H\hat{\vec{x}} &   \hat{\vec{d}}_{ZFR} &= \mat{A}^{-1}\hat{\vec{x}}
% \end{align}

% and $(\cdot)^H$ denotes Hermition conjugate.  In a QAM-modulated GFDM
% system, the pulse shaping filter $g[n]$ can be non-orthogonal and
% therefore introduce significant self-interference, that needs to be
% considered at the receiver.  The MFR ignores this interference, but
% maximizes the signal-to-noise ratio per subcarrier. On the other hand,
% the ZFR cancels this self-interference, but introduces a noise
% enhancement, depending on the bandwidth of the filter
% \cite{Matthe2014} which reduces the system performance.

% Due to the missing phase shift between subcarriers when using CR
% filters for OQAM, the application of CR-OQAM to GFDM can be written
% straight-forward by


% In a real CR-OQAM-GFDM implementation the matrix multiplication can be
% efficiently done in the frequency domain by exploiting the sparsity of
% $G(f)$ \cite{Michailow2014}.

%\subsection{Space-Time Coding for GFDM/CR-OQAM}
% In earlier works it has already been shown how OQAM can be combined
% with orthogonal space-time block codes \cite{Lele2010},
% \cite{Bellanger2008}, however at the cost of increased receiver
% complexity or specific assumptions on the channel.
Time-reversal space-time coding (TR-STC) \cite{Al-Dhahir2001} can be
applied to GFDM/CR-OQAM to provide transmit diversity in a fading
multipath environment. TR-STC was initially developed for
single-carrier systems, where two subsequent time domain codewords are
decoupled by a CP and jointly space-time encoded.  For GFDM, the idea
is to treat one block as a single-carrier codeword and apply the
TR-STC onto two subsequent GFDM blocks \cite{Michailow2014}. At the
receiver, the time-domain signal of both GFDM blocks is recovered by
the TR-STC and later on processed by the GFDM detector.  The approach
can be applied to both GFDM and GFDM/(CR-)OQAM, but GFDM/OQAM is
expected to outperform conventional GFDM due to the kept
orthogonality.


% The application of
% TR-STC to GFDM is described in detail in \cite{Matthe2014a} and is only
% shortly repeated here.  The idea is to treat one GFDM block as a
% single-carrier codeword and apply the TR-STC onto two subsequent GFDM
% blocks. Let $\vec{x}_1$ and $\vec{x}_2$ denote two subsequent CR-OQAM
% modulated GFDM blocks and let $\overleftarrow{u}=u[(-n)\mod N]$ denote the
% time-reversed version of $\vec{u}$, the transmit signal on both
% antennas at the transmitter is given by

% \begin{center}
% \begin{tabular}{l|cc}
%        & Antenna 1                & Antenna 2 \\\hline
% Time 1 & $\vec{x}_{1,1}=\vec{x}_1$ & $\vec{x}_{2,1}=-\overleftarrow{x}_2^*$      \\
% Time 2 & $\vec{x}_{1,2}=\vec{x}_2$ & $\vec{x}_{2,2}=\overleftarrow{x}_1^*$      \\
% \end{tabular}
% \end{center}

% where $\vec{x}_{i,j}$ denotes the signal that is transmitted with the
% $i$th antenna at the $j$th time instance.  After both signals have
% passed the fading multipath channel with frequency response $H_{i}$,
% where $i$ is the transmit antenna index, at the receiever the CP is
% removed from both blocks and the transmitted code words are recovered
% in the frequency domain according to
% \begin{align}
%     \hat{\vec{X}}_{1}   &= \frac{\vec{H}^*_{1}\vec{Y}_{1}   + \vec{H}_{2}\vec{Y}^*_{2} }{|\vec{H}_1|^2+|\vec{H}_2|^2}\\
%     \hat{\vec{X}}_{2} &= \frac{\vec{H}^*_{1}\vec{Y}_{1} - \vec{H}_{2}\vec{Y}^*_{2}}{|\vec{H}_1|^2+|\vec{H}_2|^2},
% \end{align}
% where $\vec{Y}_j$ is the Fourier transform of the received signal in the
% $j$th time instance and $\hat{\vec{X}}_i$ is the Fourier transform of
% $\vec{x}_i$. There, all vector operations are to be understood
% elementwise.  After the space-time combining has been carried out at
% the receiver, the CR-OQAM demodulation as described in the previous
% section is carried out on $\mathcal{F}^{-1}\{\hat{X}_i\}$.

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G.~Wunder, P.~Jung, M.~Kasparick, T.~Wild, F.~Schaich, Y.~Chen, S.~Brink,
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