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\begin{document}
\title{}
\author{%
  \IEEEauthorblockN{Sheng Yang, Laurent Mazet, and Marc de Courville}
  \IEEEauthorblockA{Motorola-Labs Paris -- Parc Les  Algorithmes, Saint Aubin -- 91193 Gif sur Yvette Cedex -- France \\
    $\{$sheng.yang, laurent.mazet, marc.de.courville$\}$@motorola.com}
}

%\thanks{}
%\IEEEspecialpapernotice{}

\maketitle

\begin{abstract}
We propose a framework of superposition coding for Multiple Input
Multiple Output~(MIMO) channels in the context of next generation
broadband systems, where the overall throughput is limited by the
achievable throughput of users near the cell edge (far users or FU).
The superposition coding scheme is proposed by Cover in
\cite{ieeeIT.cover72}. It is shown in \cite{ieeeIT.bergmans74,
  ieeeIT.bergmans74a} that superposition coding is optimal for Single
Input Single Output~(SISO) channels, i.e., it achieves the capacity
region of such channels.
\end{abstract}


\begin{IEEEkeywords}
Superpostion coding, SDMA, WIMAX, Interference management
\end{IEEEkeywords}

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

This document addresses the design of new Multi User (MU) Multiple Input Multiple Output (MIMO) schemes in the context of next generation of broadband systems. In this type of networks where coordination between base stations is not required, users in neighboring cells interfere with the transmission in a given cell. Thus, the average throughput of a cell is limited by the achievable throughput for border cell users. In \cite{ieeeIT.cover72}, Cover proposed and analyzed a solution, denoted ``superposition coding'' for reliable transmission over broadcast channel (a single source attempting to communicate information simultaneously to several receivers). Moreover,  Cover showed that superposition coding outperforms time-sharing techniques in term of throughput. This has been enforced by Bergmans studies in \cite{ieeeIT.bergmans74,ieeeIT.bergmans74a} which demonstrated the optimality of superposition coding scheme. At last, some recent works of Shamai\cite{isit.shamai97} and Liu\cite{isit.liu02} show the feasibility of superposition coding with a reduce number of coding level.

We present a framework of superposition coding for MIMO channels in the context of next generation broadband systems, where the overall throughput is limited by the achievable throughput of users near the cell edge (far users or FU). This framework is suitable for a variety of systems with different level of knowledge on CSI at the transmitter side, different level of receiver complexity, and different number of data streams for FU and NU.

\section{Signal model}
\label{sec:signal-model}

\begin{figure}[ht]
  \centering
    \includegraphics[width=\linewidth]{near_far}  
  \caption{MIMO downlink channel with one BS, one near user, and one far user.}
  \label{fig:mimo-downl-chann}
\end{figure}

The considered downlink MIMO channel is composed of one BS with $M$
transmit antennas, one near user, and one far user both of which are
equipped with $N$ receive
antennas~(Figure~\ref{fig:mimo-downl-chann}). In addition, we suppose
that $M\geq N$. The signal model is
$$\Matrix{\Yn\\\Yf} = \Matrix{\Hn\\\Hf}\mat{X} + \Matrix{\Zn\\\Zf}, $$
where $\mat{X}\in\mathbb{C}^{M\times T}$ is the transmitted signal with $T$
being the interval during which the channels remain unchanged;
$\Hn,\Hf\in\mathbb{C}^{N\times N}$ are respectively the BS-NU and BS-FU
channel matrices; $\Zn,\Zf\in\mathbb{C}^{N\times T}$ are Additive White
Gaussian Noise~(AWGN) with identical and independently
distributed~(i.i.d.)  $\mathcal{CN}(0,1)$ entries at the NU and FU
receiver sides, respectively.


\section{Proposed framework}
\label{sec:proposed-framework}

The idea of superposition coding is captured by the following form of
$\mat{X}$
$$\mat{X} = \sqrt{1-\alpha} \Xf + \sqrt{\alpha} \Xn,$$
where $\Xn,\Xf\in\mathbb{C}^{M\times T}$ are the signals intended for the NU
and FU, respectively; $\alpha$ is the superposition weight. Let us
recall that the goal of the considered superposition coding scheme is
to boost the capacity of the NU by sacrificing a negligible fraction
of the FU capacity. Therefore, $\alpha$ is directly related to the
target loss of the FU capacity and is \emph{a priori} close to $0$.


\subsection{Linear precoding scheme}
\label{sec:linear-precoding}

The first element of the proposed framework is applying the same
linear precoder to both the FU and NU signals, i.e., 
\begin{equation} \label{eq:precod}
  \Matrix{\Xn&\Xf} = \mat{T}\,\Matrix{\Sn&\Sf}
\end{equation}%
with $\mat{T}\in\mathbb{C}^{M\times M}$ being some precoding matrix that
depends on the available transmitter CSI at the BS. In particular,
when no transmitter CSI is available, we set $\mat{T}=\mat{I}$, i.e.,
no linear precoder is used. In the following, we assume that the BS
has perfect transmitter CSI on the BS-NU channel. This assumption is
motivated by the fact that the BS-NU channel is usually strong enough
for estimation. Nevertheless, the established results may be
generalized straightforwardly to the general case without this
assumption. 

Let 
$$\Hn=\Vnt\Matrix{\Sigman&\mat{0}} \Un$$
be the Singular Value Decomposition~(SVD) of $\Hn$ with $\Vn,\Un$
being respectively $N\times N$ and $M\times M$ unitary matrices and $\Sigman$
being a $N\times N$ diagonal matrix with decreasingly ordered singular
values of $\Hn$. Then, we set the precoding matrix $\mat{T}$ in
(\ref{eq:precod}) as
$$\mat{T}=\Unt.$$


\subsection{Coding and decoding strategies}
\label{sec:coding-decod-strat}


The second element of the proposed framework is assigning the set of
$q$ strongest eigen-modes of $\Hn$ to carry the NU signal and another
set of $q'$ eigen-modes of $\Hn$~(say, the $q'$ weakest eigen-modes)
to the FU signal. This choice is reflected by the following setting on
$\Sn$ and $\Sf$
\begin{equation*}
  \Sn = \Matrix{\mat{N}_{q\times T} \\ \mat{0}_{(M-q)\times T}},\quad
  \Sf = \Matrix{\mat{0}_{(M-q')\times T} \\ \mat{F}_{q'\times T}}.
\end{equation*}

With the above choice, the equivalent channels seen by the NU and FU
are
\begin{align}
  \Vn\Yn &=\sqrt{\alpha}\,\Sigmanone \mat{N} + \sqrt{1-\alpha} \Matrix{\Sigmantwo&\mat{0}}\mat{F} + \Vn\Zn \label{eq:eqch_n}\\
  \Yf &=\sqrt{1-\alpha}\, \Hftwo\, \mat{F} + \left(\sqrt{\alpha}\, \Hfone
    \mat{N} + \Zf\right) \label{eq:eqch_f}
\end{align}%
where $\Sigmanone\in\mathbb{C}^{N\times q},\Sigmantwo\in\mathbb{C}^{N\times
  (q'+N-M)}$ are respectively the first $q$ and last $q'+N-M$ columns
of the matrix $\Sigman$; $\Hfone\in\mathbb{C}^{N\times
  q},\Hftwo\in\mathbb{C}^{N\times (q')}$ are respectively the first $q$ and
last $q'$ columns of the matrix $\Hf'\triangleq\Hf\,\Unt$. Hence, the
NU sees $q$ parallel SISO channels in each one of which it is
interfered by the FU signal, while the FU sees an equivalent $q'\times N$
MIMO channel in which it is interfered by the NU signal.

The proposed coding strategy for each user is as follows. The BS
transmits
\begin{itemize}
\item $q$ independent data streams~(with or without optimal power
  allocation) to the NU, now that it sees $q$ parallel SISO channels;
\item a low-rate STBC to the FU, since the attenuation is strong and
  the FU channel is unknown at the transmitter side.
\end{itemize}
In a word, we combine the open-loop transmission to the NU with the
closed-loop transmission to the FU. The proposed decoding strategy
for each user is described below.
\begin{itemize}
\item The NU tries to decode the FU signal first, since the FU signal
  is much stronger than the NU signal due to the setting of $\alpha$.  If
  the decoding is successful, the FU signal is removed and NU proceeds
  with the decoding of its own signal. If the NU fails in decoding the
  FU signal, it will treat the FU signal as noise and decode its own
  signal.
\item The FU decodes its own signal by treating the NU signal as
  noise. Since $\alpha$ is close to $0$, the interference from the NU
  signal is small. 
\end{itemize}


\subsection{Capacity and error probability}
\label{sec:capacity-error-rate}


Let $I(\yn;\vect{f})$, $I(\yn;\vect{n})$, $I(\yf;\vect{f})$, and
$I(\yn;\vect{n}|\vect{f})$ be the mutual information between FU
transmitted signal and NU received signal, between NU transmitted
signal and NU received signal, between FU transmitted signal and FU
received signal, and between NU transmitted signal and NU received
signal conditional on FU transmitted signal, respectively. From
(\ref{eq:eqch_n}) and (\ref{eq:eqch_f}), we have
\begin{align*}
  I(\yn;\vect{f}) &= \logdet{\mat{I} + (1-\alpha)\Sigmantwo\Qf'\herm{\mat{\Sigma}}_{\text{n},2}\inv{\left(\mat{I}+\alpha\Sigmanone\Qn\herm{\mat{\Sigma}}_{\text{n},1}\right)} },\\ 
  I(\yn;\vect{n}) &= \logdet{\mat{I} + \alpha\Sigmanone\Qn\herm{\mat{\Sigma}}_{\text{n},1}\inv{\left(\mat{I}+(1-\alpha)\Sigmantwo\Qf'\herm{\mat{\Sigma}}_{\text{n},2}\right)} }, \\ 
  I(\yf;\vect{f}) &= \logdet{\mat{I} + (1-\alpha)\Hftwo\Qf\herm{\mat{H}}_{\text{f},2}\inv{\left(\mat{I}+\alpha\Hfone\Qn\herm{\mat{H}}_{\text{f},1}\right)} },\\
  I(\yn;\vect{n}|\vect{f}) &= \logdet{\mat{I} + \alpha\Sigmanone\Qn\herm{\mat{\Sigma}}_{\text{n},1} }, 
\end{align*}%
where we have assumed that Gaussian signaling has been used, i.e., the
NU and FU signals are codewords from random codebooks generated with
$\mathcal{CN}(0,\Qn)$ and $\mathcal{CN}(0,\Qf)$ distributions,
respectively; $\Qf'$ is the $(q'+N-M)\times(q'+N-M)$ leading principle
submatrix of $\Qf$. As a matter of fact, it is optimal to set $\Qn$
and $\Qf$ diagonal.  Moreover, since BS-NU channel is known to the BS,
we can optimally allocate the power $\alpha P_{\text{Tx}}$ over $q$
parallel channels. On the other hand, since no transmitter CSI is
available on the BS-FU channel, isotropic power allocation is used for
the FU, i.e.,
$$\Qf=\frac{P_{\text{Tx}}}{q'N_0}\mat{I}$$
with $N_0$ being the variance. 


Now, let $\Rn$ and $\Rf$ be the target data rates per channel use of
the NU and FU, respectively. By supposing that capacity achieving
codes are used, the error probability with the proposed decoding
scheme can approach the following outage probability
\begin{align*}
  P_{\text{out,f}}(\Rf) &= 1-\text{Prob}\left\{I(\yf;\vect{f})\leq\Rf \right\}, \\
  P_{\text{out,n}}(\Rf, \Rn) &= 1-\text{Prob}\left\{
    \{I(\yn;\vect{f})\leq\Rf,\ \text{and}\
    I(\yn;\vect{n}|\vect{f})\leq\Rn\}\ \text{or}\
    \{I(\yn;\vect{n})\leq\Rn\} \right\}.
\end{align*}%
We define the following average capacity (or outage capacity) for both
the FU and NU
\begin{align*}
  \Cf &\triangleq \max_{\Rf} \left\{\Rf\, (1-P_{\text{out,f}}(\Rf)) \right\}, \\
  \Cn &\triangleq \max_{\Rn} \left\{\Rn\, (1-P_{\text{out,f}}(\Rf^*, \Rn)) \right\},
\end{align*}%
where $\Rf^*\triangleq\arg\max_{\Rf} \left\{\Rf
  (1-P_{\text{out,f}}(\Rf))\right\}$. Note that we propose to optimize
the FU capacity and NU capacity in order, since the FU is in higer
priority than the NU in the considered scenario.


\section{Choice of superposition weight $\alpha$}
\label{sec:superp-weight-alpha}

The superposition weight $\alpha$ is chosen in accordance with a
target loss in terms of the FU capacity. Note that the loss is
supposed to be negligible, which means that $\alpha$ is close to $0$.
For simplicity, the target loss will be expressed in FU ergodic
capacity in the following, although outage capacity can be used as
well. Instead of searching for the exact value of $\alpha$, we propose
to get an approximative value via a linear approximation of the
function $I(\yf;\vect{f})$ with respect to $\alpha$. In this way,
$\alpha$ is in closed form expression. 

First, let us rewrite 
\begin{multline*}
  I(\yf;\vect{f}) = \logdet{\mat{I} +
    \Hftwo\Qf\herm{\mat{H}}_{\text{f},2}}  - \logdet{\mat{I} + \alpha\Hfone\Qn\herm{\mat{H}}_{\text{f},1}} \\
  + \logdet{\mat{I} + \alpha
    \left( \Hfone\Qn\herm{\mat{H}}_{\text{f},1} -
      \Hftwo\Qf\herm{\mat{H}}_{\text{f},2} \right) \inv{\left(\mat{I} +
      \Hftwo\Qf\herm{\mat{H}}_{\text{f},2}\right)} } 
\end{multline*}
Then, since $\alpha$ is close to $0$ and that the entries of $\Hf$ are of
small amplitude, we have 
\begin{multline*}
  I(\yf;\vect{f}) \approx \logdet{\mat{I} +
    \Hftwo\Qf\herm{\mat{H}}_{\text{f},2}} \\ - \alpha\,
  \text{trace}\left(\Hfone\Qn\herm{\mat{H}}_{\text{f},1} - \left(
      \Hfone\Qn\herm{\mat{H}}_{\text{f},1} -
      \Hftwo\Qf\herm{\mat{H}}_{\text{f},2} \right) \inv{\left(\mat{I}
        + \Hftwo\Qf\herm{\mat{H}}_{\text{f},2}\right)} \right),
\end{multline*}
where we have used the fact that
$$\logdet{\mat{I}+\mat{A}}=\text{trace}(\mat{A})+O\left((\text{trace}(\mat{A}))^2\right),$$
for all $\mat{A}$ with all real eigenvalues. Finally, we deduce that,
for a loss of $\rho\,\%$ in terms of FU ergodic capacity, we should set
\begin{equation*}
  \alpha(\rho) = (\rho\,\%) \frac{\mathbb{E}\left\{ \logdet{\mat{I} +
    \Hftwo\Qf\herm{\mat{H}}_{\text{f},2}}\right\}}{\mathbb{E}\left\{ \text{trace}\left(\Hfone\Qn\herm{\mat{H}}_{\text{f},1} - \left(
      \Hfone\Qn\herm{\mat{H}}_{\text{f},1} -
      \Hftwo\Qf\herm{\mat{H}}_{\text{f},2} \right) \inv{\left(\mat{I}
        + \Hftwo\Qf\herm{\mat{H}}_{\text{f},2}\right)} \right)\right\}},
\end{equation*}%
where the numerator stands for the FU ergodic capacity without
superposition coding. Note that the optimal $\alpha$ is proportional to
the loss $\rho$.


\section{Two examples}
\label{sec:two-examples}

For a given number of NU data streams $q$, we consider two extreme
choices of the dimension of FU signal $q'$: $q'=M$ and $q'=M-q$. The
first choice corresponds to the pure superposition coding, while the
second one corresponds to a semi-orthogonal scheme where the FU signal
is invisible to the NU.
\begin{table}[htbp]
  \centering
  \begin{tabular}[c]{|c|c|c|}
    \hline
    $q'$ & FU received signal: $\Yf$ & NU received signal: $\Vn\Yn$ \\
    \hline 
     $M$ & $\sqrt{1-\alpha}\, \Hf'\, \mat{F} + \left(\sqrt{\alpha}\, \Hfone
      \mat{N} + \Zf\right)$ & $\sqrt{\alpha}\,\Sigmanone \mat{N} + \left(\sqrt{1-\alpha}\, \Sigman\mat{F}' + \Vn\Zn\right)$ \\\hline
     $M-q$ & $\sqrt{1-\alpha}\, {\Gfone}\, \mat{F} + \left(\sqrt{\alpha}\, \Hfone
      \mat{N} + \Zf\right)$ & $\sqrt{\alpha}\,\Sigmanone \mat{N} +  \Vn\Zn$ \\\hline
  \end{tabular}
  \caption{Two examples of the proposed framework where $\mat{F}'$ is the first $N$ rows of $\mat{F}$, and ${\Gfone}$ is the $M-q$ last columns of $\Hf'$.}
  \label{tab:two-example-proposed}
\end{table}

As shown in Table~\ref{tab:two-example-proposed}, by switching $q'$
from $M$ to $M-q$, the NU does not see the interference from the FU
signal. It means that the NU does not need to perform SIC to retrieve
its own signal. The superposition coding is transparent to both the NU
and FU with this setting. On the other hand, for the FU, the
equivalent channel becomes a $(M-q)\times N$ MIMO channel instead of a $M\times
N$ MIMO channel. Therefore, setting $q'=M-q$ trades the FU performance
for the reduction of complexity at the NU receiver side. 

\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{avgCapa_4Tx_2Rx_1WF_0OSTBC_001loss}
  \caption{Outage capacity versus BS-NU distance.}
  \label{fig:outage-capac-vers}
\end{figure}

In Figure~\ref{fig:outage-capac-vers}, the numerical results on the
outage capacity versus the BS-NU distance are illustrated. There, we
consider a single cell setting with $M=4$ and $N=2$. The FU is located
at $750$ meters to the BS, while the BS-NU distance varies from $100$
to $350$ meters. All the channel coefficients are i.i.d.~Rayleigh
distributed. The pathloss factor is $3.5$. The target loss of FU
ergodic capacity is fixed to $1\%$. Two choices on the number of NU data
streams are considered: $q=1$ and $2$. Waterfilling power allocation is
applied to the NU signal when $q=2$. Accordingly, $q'$ is set to $M$
and $M-q$ for both values of $q$.  It is shown in the figure that the
FU outage capacity depends only on the value of $q'$. Setting $q'=2$
incurs roughly $25\%$ of FU capacity loss as compared to the case with
$q'=4$. As for the NU capacity, setting $(q,q')=(2,2)$ provides better
performance than the other settings. This is due to the fact that
setting $q'=2$ results in a larger $\alpha$ than the other setting for a
given FU capacity loss. It should also be noted that using single NU
data stream~($q=1$) has similar performance as using double NU data
streams~($q=2$) in terms of the NU capacity for BS-NU distance larger
than $200$ meters. This is because waterfilling tends to allocate all
the power to a single stream as the channel quality degrades. The case
without any transmitter CSI is also studied for completeness. In this
case, we have $\mat{T}=\mat{I}$, $(q,q')=(4,4)$. Both the NU and FU
signals have isotropic power allocation. The loss of the no
transmitter CSI case to the other cases shows the advantage of the
exploitation of transmitter CSI.




\section{Incorporation of orthogonal STBC}
\label{sec:incorp-orth-stbc}

The above capacity analyses are based on the assumption of Gaussian
signaling and provide an upper bound on the data rates of both FU and
NU. However, in order to realize the promised data rates, capacity
achieving codes are needed. For the NU, sophisticated turbo codes or
LDPC codes exist, since the equivalent channels are SISO channels.
Efficient decoding schemes can also be found. For the FU, on the other
hand, practical low-rate space-time precoders should be used to combat
channel fading with the absence of transmitter CSI. We adopt the
orthogonal STBCs proposed by Tarokh \emph{et
  al.}~\cite{ieeeIT.tarokh99}, since 1)~they are of low rate, and
2)~the decoding complexity is as low as in the SISO case. The maximum
achievable rate of orthogonal design is $1$ symbol per channel use
with Alamouti code for $q'=2$, and $\frac{3}{4}$ symbol per channel
use for $q'=3,4$. With these OSTBC precoders, numerical results with
the same system configuration as the case without OSTBC precoders are
shown in Figure~\ref{fig:outage-capac-vers2}. Unlike the results in
Figure~\ref{fig:outage-capac-vers}, the setting $(q,q')=(2,2)$
outperforms the other settings in terms of the FU capacity. This is
because setting $q'=2$ means that Alamouti code~(rate $1$) can be used
while setting $q'>2$ leaves the BS no choice but using codes with
lower rate~(rate-$\frac{3}{4}$). Also note that with Alamouti code,
the most part of the promised FU capacity in
Figure~\ref{fig:outage-capac-vers} is realized, while the use of codes
in higher dimensions~($q'=3,4$) incurs a large loss in terms of the FU
capacity. As for the NU capacity, it depends uniquely on the value of
$q$ and is independent of $q'$. In a word, the setting $(q,q')=(2,2)$
gives both better capacity performance and lower receiver complexity
than the other settings, provided that OSTBCs are used for FU.


\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{avgCapa_4Tx_2Rx_1WF_1OSTBC_001loss}
  \caption{Outage capacity versus BS-NU distance, OSTBC is used by FU.}
  \label{fig:outage-capac-vers2}
\end{figure}

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

The technology presented in this document has been developed in the
framework of the B4G-Coope\-ra\-tive project which intends to define next
generation of broadband wireless systems intended to fulfill the
IMT-Advanced requirements.

The intend is to construct a proper portfolio in order to address
incoming standards such as LTE+ or IEEE802.16m that will compete for
spectrum certification access to the frequency bands to be allocated
by WRC07. The target date for standard approval is around 2009 with
earlier communication for adoption.

Superposition coding schemes will be pushed by Motorola in the
IEEE802.16m standard starting in the March 2008 meeting on the basics
of the concept and disclosing technical details of the proposed method
in the May 2008 meeting related to the MIMO scheme impact on the frame
structure discussions.

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