\documentclass[a4paper,12pt]{letter}

\usepackage{amsmath,amssymb,amsfonts,mathrsfs,amscd,amsthm}

\name{Laurent Mazet \\
  Sheng Yang \\
  Lina Mroueh \\
  Marc de Courville}
\signature{L. Mazet, S. Yang, L. Mroueh and M. de Courville}
\address{Laurent Mazet \\
  17, rue d'Al\'esia \\
  75014 Paris \\
  FRANCE}
\location{Paris}

\newcommand{\paragraph}[1]{\textbf{#1}}
\newcommand{\comment}[1]{\begin{quote}#1\end{quote}\smallskip}

\begin{document}

\begin{letter}{
    Prof. Nevio Benvenuto \\
    Editor \\
    IEEE Transactions on Communications}
  \opening{Dear Professor, dear Reviewers,}
We would like to thank you for your suggestions and comments regarding our manuscript entitled ``\emph{Non-Linear Precoding with Periodically Flipped Constellation for MIMO Downlink Channels}'' (denoted TCOM-TPS-08-0652). We also appreciate your remarks bringing to our attention for valuable recommendation that improves the presentation of our work. Please see below for details. 

\paragraph{Editor Comments}
\begin{enumerate}

\item  \comment{State your channel model, An OFDM with 10 subcarriers is no sense, try a standard configuration}

We agree with the editor that the simulations parameters as described in the first version of this manuscript are not very clear. 

In our simulation, the scenario is detailed in the following. We assume a single cell with base station having $M = 4$ antennas, and $K$ single antennas users ($N = 1$). In this case, it is well known for a MIMO BC that the maximal total multiplexing gain that can  be achieved is $r = \min(M,KN) = 4$  when $K$ is large. For this end, a scheduling algorithm (such as Proportional Fair scheduling) is used in the cell, where $4$ users are selected.  

A MIMO OFDMA system, having $N$ subcarriers, where $N = 512$, $1024$, or $2048$ in practical standards is used. On each subcarrier, the channel is considered flat. 
Knowing that channel coefficients on two subcarriers $f_i$ and $f_j$ such that $|f_i - f_j| \geq B_c$ ($B_c$ being the coherence band of the channel) are uncorrelated, a frequency interleaver can be used to have independent channel on interleaved subcarriers.
For each user, a cluster of $10$ subcarriers is assigned. Note that here $10$ refers to the number of subcarriers assigned to each user and not to the OFDM itself.  

The uncoded performance correponds to the case when no outer code is used (the convolutional code), and the coded modulation corresponds to the case with a convolutional code $[$o133~o171$]$.

We believe that the simulation context was not sufficiently detailed in the first version. Changes are made in the revised version (III-C). 

\item \comment{A target of BER= $10^{-1}$  for comparison is very high!}

For uncoded performance in fig. 3, gains are observed only in the low SNR regime. 

This can be explained by the fact that the PFC enhances performance of sphere encoding against noise perturbation. At low SNR, the probability that the error is due to noise perturbation is significant. However, at high SNR, the variance of the noise is very small, this error probability is reduced.  

For the coded performance in fig. 4,  using convolutional code allows us to operate at low SNR regime but with a higher BER of $10^{-2}$. A gain of $1.5$ dB is obtained. 

Clarifications about this point are given in the revised version(III-C).
\end{enumerate}

\paragraph{Reviewer 1}
\begin{enumerate}

\item \comment{Typos errors} 

All typos errors have been changed in the revised document as suggested.

\item \comment{Avoid to split the Schnorr Euchner in two parts}

The Schnorr Euchner is splitted in two parts as it was not possible that the whole algorithm fits in one column for a latex compilation with two columns. That's why, we splitted into two parts. For the sake of clarity, (part II) has been replaced by (Algorithm 1 cont'd) in the revised version. 

\item \comment{Comparison between PFC with the only standard encoding is reductive. It can be improved by extending the comparison using further different schemes to show the effective benefit of PFC.}

Actually, in the MIMO Broadcast Channel literature, a large number of precoder is suggested: Regularized Precoding schemes (MMSE), Channel inversion... The sphere encoding is proposed in order to limit the power at the transmitter side, and therefore enhance the error probability performance at high SNR. 
In our contribution, we propose to use the Periodically Flipped constellation instead of the replicated ones to have additional gains compared to sphere encoding in the low SNR regime. These periodically Flipped Constellations can be used with other precoding schemes without loss of generality. There is no restriction on the use of this strategy to the channel inversion case. 

For example, for the regularized channel inversion, by using an adequate regularization parameter, performance can be largely improved compared to channel inversion. 
However as shown in the part I and II of the original sphere encoding paper, this is not the case for the regularized sphere encoding compared to the sphere encoding with channel inversion.  There is only a small improvement in term of error performance at low SNR as well as at high SNR. That's why, in this case we would rather use PFC for a reduced complexity with channel inversion rather than with the regularized one. 

A more details on that has been added to the paper in section III part C.
\item \comment{Abstract specify that the better error resistance of PFC is limited to small constellation and not for low error rate scenarios ($< 10{^-2}$).}

We have changed the abstract as suggested. 
\end{enumerate}

\paragraph{Reviewer 2}
\begin{enumerate}

\item \comment{The introduction is more a very extended abstract than an introduction} 

We agree with the reviewer on details that should not be mentioned so early in the paper. These details has been provided in an easiest manner in the revised manuscript. 

\item \comment{The article could be shortened since the sphere decoder has already been described in so many articles.}

We do not provide a full description of the Sphere Encoder but, only what is necessary to detail modifications need by our encoding scheme. In the revised version, some of these details are deleted. Some others details are essential for the consistency of the modified version of Schnorr Euchner. By the way, we want to provide in this paper the concept of the PFC and a way to implement it. 

\item \comment{A similar technique using a unique sphere decoder onto different cossets is also described in Decoding the golden space-time trellis coded modulation published in communication letters issue 8 year 2008, see if a comparison or a citation is maybe needed.}

In the referred paper, the main objective is to design a unique decoder for $E_8 = 2\mathbb{Z}^8 + [8\;4\;4]$. 

The basic decoder consists to choose the point having the minimal distance among the minimal $16$ distance computed in cosssets. The unique decoder operates in $\mathbb{Z}^8$ while choosing only points that belongs to $E_8$, while skipping other points. It is also shown that complexity of this new decoder is reduced compared to the compositions of the $16$ others sphere decoders.

The idea of having a unique decoder instead of having a composition of decoder can also apply to our case. As presented in the paper instead of using a decomposition of sphere encoder as shown by solution 1 in IV-C, a unique sphere decoder is proposed in solution 2.  
In the revised version, we mention briefly this in section V part C.
 
\end{enumerate}

Thank you very much for your attention and consideration. We look
forward to hearing your kind decision.

  \closing{Sincerely,}

  \end{letter}

\end{document}
