Changeset 255 in ETALON for reconstruction


Ignore:
Timestamp:
Aug 3, 2015, 2:10:00 AM (9 years ago)
Author:
delerue
Message:

Paper updated

Location:
reconstruction/long_paper3
Files:
5 edited

Legend:

Unmodified
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Removed

  • reconstruction/long_paper3/phase_reconstruction_paper.aux

    r248 r255  
    1111\@writefile{toc}{\contentsline {section}{Reconstruction methods}{1}}
    1212\@writefile{toc}{\contentsline {section}{Description of the simulations}{1}}
    13 \citation{E203prstab}
    14 \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Example of profiles giving very different $\chi ^2$ despite being relatively similar. $\chi ^2_{SN}=3.8219e-08, \chi ^2_{O}=7.2661e-08$; For profile with sine noise: FW0.1M=0.0241, FW0.2M=0.044 FWHM=0.0621 FW0.8M=0.1849 FW0.9M=0.3619. For offset profile all FWXM=0. \relax }}{2}}
     13\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Example of profiles giving very different $\chi ^2$ despite being relatively similar.   $\chi ^2_{\unhbox \voidb@x \hbox {sine noise}}=3.8219e-08, \chi ^2_{\unhbox \voidb@x \hbox {offset}}=7.2661e-08$; For profile with sine noise: FW0.1M=0.0241, FW0.2M=0.044 FWHM=0.0621 FW0.8M=0.1849 FW0.9M=0.3619. For offset profile all FWXM=0. \relax }}{2}}
    1514\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
    1615\newlabel{Offsine}{{1}{2}}
    1716\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Effect of scaling the constraints on the parameters $\sigma _i$ (top) and $\mu _i$ (bottom) on the $\chi ^2$.\relax }}{2}}
    1817\newlabel{sigma_chi2}{{2}{2}}
     18\citation{E203prstab}
    1919\newlabel{eq:lamb}{{7}{3}}
    20 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Comparison of different samplings with $\chi ^2$ criterium (top) and $\Delta _{FWHM}$ (bottom)\relax }}{3}}
     20\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Comparison of different samplings with $\chi ^2$ criterium (top) and $\Delta _{FWHM}$ (bottom).\relax }}{3}}
    2121\newlabel{samp}{{3}{3}}
    22 \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.\relax }}{3}}
     22\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Detector position for linear sampling for a minimal detector distance of $5^o$ (top) and $10^o$ (bottom) .\relax }}{3}}
    2323\newlabel{lin12}{{4}{3}}
    2424\citation{pchip}
    2525\citation{VBthesis}
    2626\citation{VBthesis,DESYthesis}
    27 \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Comparison of different sampling with number of MDD with $\chi ^2$ criterium (top) and $\Delta _{FWHM}$ (bottom). Ls is linear sampling with $1^o,5^o,10^0$ MDD and Ts is Triple sine sampling; mx mean the reconstruction use th maximum number of detectors (blue and red dots on figure \ref  {lin12})\relax }}{4}}
     27\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Comparison of different sampling with number of MDD with $\chi ^2$ criterium (top) and $\Delta _{FWHM}$ (bottom). Ls is linear sampling with $1^o,5^o,10^0$ MDD and Ts is Triple sine sampling; mx mean the reconstruction use the maximum number of detectors (blue and red dots on figure \ref  {lin12}).\relax }}{4}}
    2828\newlabel{biglin}{{5}{4}}
    2929\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Effect of the sampling frequencies on the $\chi ^2$ (top) and $\Delta _{FWHM}$ (bottom). \relax }}{4}}
    … …  
    4242\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Comparison of different HF extrapolation for Lorenzians~:histogram with mean $\chi ^2$ (top) and $\Delta _{FWHM}$ (bottom).\relax }}{6}}
    4343\newlabel{hf3}{{11}{6}}
    44 \bibcite{OTR_LURE}{1}
    45 \bibcite{ODR_Cianchi}{2}
    4644\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Examples of well reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{7}}
    4745\newlabel{good_profiles}{{12}{7}}
    48 \@writefile{toc}{\contentsline {section}{Discussion}{7}}
    4946\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Example of poorly reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{7}}
    5047\newlabel{bad_profiles}{{13}{7}}
    51 \bibcite{Doucas_ESB}{3}
    52 \bibcite{KK}{4}
    53 \bibcite{E203prstab}{5}
    54 \bibcite{Pelliccia:2014vba}{6}
    55 \bibcite{VBthesis}{7}
    56 \bibcite{pchip}{8}
    57 \bibcite{LaiS}{9}
    58 \bibcite{DESYthesis}{10}
    59 \@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces {$\Delta _{FWHM}$ (top) and $\chi ^2$ (bottom) distribution of 1000 simulations reconstructed using the Hilbert transform method (black line) and Kramers-Kronig reconstruction method (red line). XXX If the top figure is delta FWHM, then the title should say so XXX }\relax }}{8}}
     48\@writefile{toc}{\contentsline {section}{Discussion}{7}}
     49\@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces {$\Delta _{FWHM}$ (top) and $\chi ^2$ (bottom) distribution of 1000 simulations reconstructed using the Hilbert transform method (black line) and Kramers-Kronig reconstruction method (red line). }\relax }}{8}}
    6050\newlabel{profiles_stats_hilbert}{{14}{8}}
    6151\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Example of reconstructed profile with zooms on the peak and tails. One can see that the profile reconstructed using the Kramers-Kronig method has a negative component. This will dominate the final $\chi ^2$ and explains why the $\chi ^2$ obtained by this method is higher as shown on figure~\ref  {profiles_stats_hilbert}.\relax }}{8}}
    … …  
    6959\@writefile{lof}{\contentsline {figure}{\numberline {19}{\ignorespaces Mean $\chi ^2$ and $\Delta _{FWXM}$ as function of noise amplitude.\relax }}{9}}
    7060\newlabel{noise}{{19}{9}}
     61\bibcite{OTR_LURE}{1}
     62\bibcite{ODR_Cianchi}{2}
     63\bibcite{Doucas_ESB}{3}
     64\bibcite{KK}{4}
     65\bibcite{E203prstab}{5}
     66\bibcite{Pelliccia:2014vba}{6}
     67\bibcite{VBthesis}{7}
     68\bibcite{pchip}{8}
     69\bibcite{LaiS}{9}
     70\bibcite{DESYthesis}{10}

  • reconstruction/long_paper3/phase_reconstruction_paper.log

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  • reconstruction/long_paper3/phase_reconstruction_paper.tex

    r248 r255  
    4141%\AddToShipoutPictureFG*{\AtTextLowerLeft{\textcolor{red}{COPYRIGHTSPACE}}}
    4242
     43%ND 20150802
     44
    4345\begin{document}
    4446%\linenumbers
    … …  
    6769where $I(\lambda)$ is the emitted intensity as a function of the wavelength.  $I_1(\lambda)$ is the intensity of the signal emitted by a single particle and $F(\lambda)$ is a form factor that encodes the longitudinal and transverse shape of the particle bunch. Recovering the longitudinal profile requires to invert this equation however this is not straightforward as the information about the phase of the form factor can not be measured and therefore is not available.
    6870
    69 A phase reconstruction algorithm must therefore be used to recover this phase. Several methods exist (see for example~\cite{KK}). Was implemented two of these methods and compared their performances below.
     71A phase reconstruction algorithm must therefore be used to recover this phase. Several methods exist (see for example~\cite{KK}).
     72In this article we describe how we implemented two of these methods and compared their performances.
    7073
    7174\section{Reconstruction methods}
    … …  
    99102\end{equation}
    100103the calculation of phase can use optimised FFT code and is much faster than  calculating the Kramers-Kronig's integral.}%VH add text
    101 Was implemented in Matlab these two different phase reconstruction methods. The Hilbert transform method has the advantage of being directly implemented in Matlab, allowing a much faster computing.
     104We implemented in Matlab these two different phase reconstruction methods. The Hilbert transform method has the advantage of being directly implemented in Matlab, allowing a much faster computing.
    102105
    103106\section{Description of the simulations}
    104107
    105 To test the performance of these methods was created a small Monte-Carlo program that randomly simulates profiles (${\cal G} (x)$) made of the combination of 5 gaussians according to the formula $ {\cal G} (x)= \sum_{i=1}^{5}  A_i  \exp{\frac{-(\frac{x}{mX} - \mu_i)^2 }{2 \sigma^2_i}} $ where $mX=2^{16}$ and $A_i$, $\mu_i$ and $\sigma_i$ are random numbers with  $x \in [1;mX]$, $A_i \in [0;1] $, $\mu_i \in 0.5 + [ -11.44 ; +11.44  ] \times 10^{-9} $ and  $\sigma_i \in [3;9] \times 10^{-9}$ . {The values of these ranges have been chosen to generate profiles that are not disconnected (that is profiles whose intensity drops to almost zero between two peaks) without being perfect gaussian.
     108To test the performance of these methods we created a small Monte-Carlo program that randomly simulates profiles (${\cal G} (x)$) made of the combination of 5 gaussians according to the formula $ {\cal G} (x)= \sum_{i=1}^{5}  A_i  \exp{\frac{-(\frac{x}{mX} - \mu_i)^2 }{2 \sigma^2_i}} $ where $mX=2^{16}$ and $A_i$, $\mu_i$ and $\sigma_i$ are random numbers with  $x \in [1;mX]$, $A_i \in [0;1] $, $\mu_i \in 0.5 + [ -11.44 ; +11.44  ] \times 10^{-9} $ and  $\sigma_i \in [3;9] \times 10^{-9}$ . {The values of these ranges have been chosen to generate profiles that are not disconnected (that is profiles whose intensity drops to almost zero between two peaks) without being perfect gaussian.
    106109% So the value of $\mu_i$ must be in the same order and approximately equal  $\sigma_i$(if distance between to peaks is less than $\sigma_i$, it can not be separated, if greater -- a lot of disjoint peaks can happen).
    107 Was checked that our conclusions are valid across this range. }
     110We checked that our conclusions are valid across this range. }
    108111% VH add and change text
    109112
    110113
    111 Using this formula was generated 1000 profiles, then was taken the absolute value of their Fourier transform $ {\cal F} = \| \mbox{FFT} \left( {\cal G}\right) \|$ and  sampled at a limited number of frequency points ($F_i = {\cal F}(\omega_i)$) as would be done with a real experiment in which the number of measurement points is limited (limited number of detectors or limited number of scanning steps).
    112 
    113 To estimate the performance of the reconstruction several estimators are available. Was choosed to use the $\chi^2$, defined as follow:
     114Using this formula we generated 1000 profiles, then took the absolute value of their Fourier transform $ {\cal F} = \| \mbox{FFT} \left( {\cal G}\right) \|$ and  sampled at a limited number of frequency points ($F_i = {\cal F}(\omega_i)$) as would be done with a real experiment in which the number of measurement points is limited (limited number of detectors or limited number of scanning steps).
     115
     116To estimate the performance of the reconstruction several estimators are available. We choose to use the $\chi^2$, defined as follow:
    114117\begin{equation}
    115118\chi^2=\sum_i\omega_i^2(O_i-E_i)^2/N,
    116119\end{equation}
    117 where $O_i$ is the observed value , $E_i$ is the expected value, $\omega_i=1/\sqrt{O_i+E_i}$ is the weight of the point, N is the number of points.\par
     120where $O_i$ is the observed value , $E_i$ is the expected (simulated) value, $\omega_i=1/\sqrt{O_i+E_i}$ is the weight of the point, N is the number of points.\par
    118121However two very similar profiles but with a slight offset, will give a worse $\chi^2$ than a profile with oscillations (see figure \ref{Offsine}). This can be partly mitigated (in the case of horizontal offset) by offsetting one profile with respect to the other until the $\chi^2$ is minimized.
    119122
    120123
    121 Also was decided to look at the FWHM which was generalized as FWXM where $X \in [0.1 ; 0.9]$ is the fraction of the maximum value at which was calculated the full width of the reconstructed profile. Here two profiles that are similar but slightly offset (in position or amplitude) will nevertheless return good values of $\chi^2$(as needed). Was created an estimator $\Delta_{FWXM}$ defined as follow:
     124Also we decided to look at the FWHM which was generalized as FWXM where $X \in [0.1 ; 0.9]$ is the fraction of the maximum value at which  the full width of the reconstructed profile was calculated (with this definition the standard FWHM is noted FW0.5M).  We created an estimator $\Delta_{FWXM}$ defined as follow:
    122125
    123126%$$
    … …  
    125128%$$
    126129\begin{equation}
    127 \Delta_{FWXM} = \left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right|
    128 \end{equation}
    129 where $\mbox{rset} = \{ 0.1 ; 0.2 ; 0.5 ; 0.8 ; 0.9\}$, $FWXM_{\mbox{orig}}$ and $FWXM_{\mbox{reco}}$ are the FWXM of the original and reconstructed profiles respectively.
     130\Delta_{FWXM} = \left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right|_{X \in  \{ 0.1 ; 0.2 ; 0.5 ; 0.8 ; 0.9\} }
     131%\Delta_{FWXM} = \sum_{X \in  \{ 0.1 ; 0.2 ; 0.5 ; 0.8 ; 0.9\} } \left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right|
     132\end{equation}
     133where
     134%$\mbox{rset} = \{ 0.1 ; 0.2 ; 0.5 ; 0.8 ; 0.9\}$,
     135$FWXM_{\mbox{orig}}$ and $FWXM_{\mbox{reco}}$ are the FWXM of the original and reconstructed profiles respectively.
     136
     137
     138Here two profiles that are similar but slightly offset (in position or amplitude) will nevertheless return good values of this estimator despite returning a rather large$\chi^2$.
    130139
    131140\begin{figure}[htbp]
    132141 \centering
    133142  \includegraphics*[width=70mm]{rev1/chiexp.eps}
    134   \caption{Example of profiles giving very different $\chi^2$ despite being relatively similar.
    135   $\chi^2_{SN}=3.8219e-08, \chi^2_{O}=7.2661e-08$; For profile with sine noise: FW0.1M=0.0241, FW0.2M=0.044 FWHM=0.0621 FW0.8M=0.1849 FW0.9M=0.3619. For offset profile all FWXM=0. }% VH change name of picture and unite with other
     143  \caption{Example of profiles giving very different $\chi^2$ despite being relatively similar. \\
     144  $\chi^2_{\mbox{sine noise}}=3.8219e-08, \chi^2_{\mbox{offset}}=7.2661e-08$; For profile with sine noise: FW0.1M=0.0241, FW0.2M=0.044 FWHM=0.0621 FW0.8M=0.1849 FW0.9M=0.3619. For offset profile all FWXM=0. }% VH change name of picture and unite with other
    136145   \label{Offsine}
    137146\end{figure}
     147
     148
     149
    138150To ensure that the choice of the parameters $\sigma_i$ and $\mu_i$ for the simulations does not biais significantly the results, their value has been varied and this is shown on figure~\ref{sigma_chi2}.
    139151
    … …  
    152164
    153165Different distributions have been used for the frequencies $\omega_i$: linear, logarithmic, triple-sine.
    154 In most sampling schemes 33 frequencies were used to make it comparable with Triple-sine. More detail about each:
     166In most sampling schemes 33 frequencies were used to make it comparable with Triple-sine. These sampling scheme are defined as follow:
    155167\begin{itemize}
    156168\item \textit{Triple-sine} This sampling matches that of the E-203 experiment at FACET~\cite{E203prstab}. Eleven detectors are located every $10^o$ around the interaction point and 3 different sets of wavelengths are used, giving the following distribution:
    … …  
    158170\frac{c}{\omega_i}=l_n (1-cos(\Theta_i ))
    159171\end{equation}
    160 with $l_n =50, 250, 1500 \mu m$ and $\Theta$ varying between $40^o$ and $140^o$.
    161 Uniform location of detectors in space corresponds to the inhomogeneous sample frequency and vice versa. %So next sampling is linear in frequecy.
    162 \item \textit{Linear sampling} There sampling points distributed uniformly. Fist and last points of sampling is first ($\omega_0$) and last ($\omega_f$) points  in Triple-sine sampling. To get sapmling frequencies, was used next formula:
    163 \begin{equation}
    164 \omega_i=\omega_0+(\omega_f-\omega_0)/32\times(0:32).
    165 \end{equation}
    166 \item \textit{Logarithmic sampling}. Point is distributed according logarithmic low. For this, was used:
    167 \begin{equation}
    168 \omega_0*exp(log(\omega_f/\omega_0)\times(0:32)/32).
    169 \end{equation}
    170 For this sampling first and last points is the same as in Triple-sine sampling. This was done to avoid impact of extrapolations on result.
     172with $l_n =50, 250, 1500 \mu m$ and $\Theta$ varying between $40^o$ and $140^o$ by steps of $10^o$.
     173%Uniform location of detectors in space corresponds to the inhomogeneous sampling frequency and vice versa. We cho
     174%So next sampling is linear in frequecy.
     175\item \textit{Linear sampling} Here sampling points are distributed uniformly. The first and last points of sampling are the first ($\omega_0$) and last ($\omega_f$) points used in the Triple-sine sampling. The following formula gives the sampling frequencies:
     176\begin{equation}
     177\omega_i=\omega_0+(\omega_f-\omega_0)/32\times[0:1:32].
     178\end{equation}
     179\item \textit{Logarithmic sampling}. Here sampling points are distributed  logarithmically:
     180\begin{equation}
     181\omega_0*exp(log(\omega_f/\omega_0)\times[0:1:32]/32).
     182\end{equation}
     183For this sampling also the first and last points are the same as in Triple-sine sampling. %This was done to avoid impact of extrapolations on result.
    171184%Further we will see, that due to space limitations of detector size, only Triple-sine sapmling is physical, but this study will give us information which sampling and further detector position are preferable.
    172185\end{itemize}
    173 {The study of the sampling is important, as it show  best position of the detectors and also  how to optimize the system. Linearly sampled spectrum gives the best result as shown on figure~\ref{samp}.
     186{The study of the sampling is important, as it shows the  best position of the detectors and also  how to optimize the system. Linearly sampled spectrum gives the best result as shown on figure~\ref{samp}.
    174187%XXX WE need to discuss this XXX  This is not surprising, because in the process of profile recovery is present  interpolation procedure  for spectrum, which is well known works best with a uniform sampling. 
    175188}
    … …  
    180193    \includegraphics*[width=70mm]{rev1/5.eps}
    181194  %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps}
    182   \caption{Comparison of different samplings with $\chi^2$ criterium (top) and $\Delta_{FWHM}$ (bottom)}%VH add  picture
     195  \caption{Comparison of different samplings with $\chi^2$ criterium (top) and $\Delta_{FWHM}$ (bottom).}%VH add  picture
    183196   \label{samp}
    184197\end{figure}
    185 {However, the linear sampling is the ideal case. In most cases the detectors have spatial dimensions (10 degrees in the case of E-203) and there is also a limit on the start and end points of detectors location (35-145 degrees for E-203). So linear sampling at a wide range of frequencies is impossible with this number of points. An investigation of how many linear sampling points can be used for a given angle difference between detectors shows that such sampling constrains strongly the number of detectors that can be used.
     198{However, the linear sampling may not be practical to realize in a the real world. One needs to take into account the spatial size of the detectors (about 10 degrees in the case of E-203) and there is also a limit on the start and end points of detectors location (35-145 degrees for E-203). So linear sampling at a wide range of frequencies is impossible with this number of points. An investigation of how many linear sampling points can be used for a given angle difference between detectors shows that such physical constraints reduce strongly the number of detectors that can be used.
    186199%For angle calculation and applying condition for first and final point was used  formula ~\ref{eq:lamb}.
    187 On figure~\ref{lin12} examples of detector positions are shown. The position of the red points is calculated using formula ~\ref{eq:lamb} and the blue are the possible positions of detector which does not break minimum detector distance (MDD) XXX This is incomplete XXX.}
     200Figure~\ref{lin12} shows examples of detector positions. The position of the red points is calculated using formula~\ref{eq:lamb} and the blue are the possible positions of detector which does not break the minimum detector distance (MDD) given on top of each plot.}
    188201\begin{figure}[htbp]
    189202 \centering
    … …  
    191204    \includegraphics*[width=70mm]{rev1/mmd10.eps}
    192205  %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps}
    193   \caption{Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.}%VH add  picture
     206  \caption{Detector position for linear sampling for a minimal detector distance of $5^o$ (top) and $10^o$ (bottom) .}%VH add  picture
    194207   \label{lin12}
    195208\end{figure}
    196209
    197 \textbf{Figure~\ref{biglin} shows a comparison of the performances achieved with such positioning for different MDD. In each case the triple sine sampling (Ts) is better than the linear sampling (Ls) and close from the maximum linear sampling (Lsmx).
     210Figure~\ref{biglin} shows a comparison of the performances achieved with such positioning for different MDD. In each case the triple sine sampling (Ts) is better than the linear sampling (Ls) and close from the maximum linear sampling (Lsmx).
    198211%As the Lsmx configuration is physically impossible,
    199 So the Ts configuration is favored and will be used in the rest of this paper. The comparison between Ts1, Ts5 and Ts10 shows that reconstruction performances are limited by the MDD.}
     212So the Ts configuration is favored and will be used in the rest of this paper. The comparison between Ts1, Ts5 and Ts10 shows that reconstruction performances are limited by the MDD.
    200213
    201214\begin{figure}[htbp]
    … …  
    203216  \includegraphics*[width=90mm]{rev1/hist1.eps} \\
    204217  \includegraphics*[width=90mm]{rev1/hist1fw.eps}
    205   \caption{Comparison of different sampling with number of MDD  with $\chi^2$ criterium (top) and $\Delta_{FWHM}$ (bottom). Ls is linear sampling with $1^o,5^o,10^0$ MDD and Ts is Triple sine sampling; mx mean the reconstruction use th maximum number of detectors (blue and red dots on figure \ref{lin12})}%VH add  picture
     218  \caption{Comparison of different sampling with number of MDD  with $\chi^2$ criterium (top) and $\Delta_{FWHM}$ (bottom). Ls is linear sampling with $1^o,5^o,10^0$ MDD and Ts is Triple sine sampling; mx mean the reconstruction use the maximum number of detectors (blue and red dots on figure \ref{lin12}).}%VH add  picture
    206219   \label{biglin}
    207220\end{figure}
    … …  
    209222
    210223%This section has been moved from elsewhere
    211 The choice of 33 frequencies for the sampling of the spectrum was made to match the current layout used on E-203. However it is important to check if there is an optimum value. Using the same simulations was used the same spectrum but sampled  with 3 to 140  points. The effect of changing the sampling frequencies on the  $\chi^2$ is shown on figure~\ref{sampling_chi2}. This study uses Triple sine sampling with 1000 profiles for each point and both reconstruction method.
     224The choice of 33 frequencies for the sampling of the spectrum was made to match the current layout used on E-203. However it is important to check if there is an optimum value. To perform this check we used the same simulations and the same simulated spectrum but sampled  with 3 to 140  points. The effect of changing the sampling frequencies on the  $\chi^2$ is shown on figure~\ref{sampling_chi2}. This study uses Triple sine sampling with 1000 profiles for each point and both reconstruction method.
    212225
    213226
    … …  
    220233   \label{sampling_chi2}
    221234\end{figure}
    222 It can be seen at figure \ref{sampling_chi2} that beyond 33 sampling points the gain on the reconstructed $\chi^2$ is marginal.
    223 
    224 
    225 %%%% Extrapolation / inerpolations
    226 
    227 After applying the sampling procedure the data need to be interpolated and extrapolated to have a larger number of points in the spectrum. Interpolation is done using Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)~\cite{pchip}, as suggested in \cite{VBthesis}.
     235
     236It can be seen at figure \ref{sampling_chi2} that beyond about 33 sampling points the gain on the reconstructed $\chi^2$ is marginal.
     237
     238
     239%%%% Extrapolation / interpolations
     240
     241After applying the sampling procedure the data need to be interpolated and extrapolated to have a larger number of points in the spectrum. Interpolation is done using Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)~\cite{pchip}, as suggested in~\cite{VBthesis}.
    228242The interpolation function must satisfy the following criteria: it must conserve the slope at the two endpoints (to have a continuous derivative) and respects monotonicity. PCHIP interpolation has been chosen as it matches these requirements.
    229243
    230244For low frequency extrapolation two methods have been investigated: Gaussian or Taylorian.
    231245
    232 In the Gaussian method, was defined the extrapolation as follow:
     246In the Gaussian method, we defined the extrapolation as follow:
    233247\begin{equation}
    234248\rho_{LF}(\omega)=Ae^{-(\omega-B)^2/2C^2}
    … …  
    255269 
    256270
    257 Approximation to the 4th order gives the following LF extrapolation:
     271Approximation to the 4th order gives the following low frequency (LF) extrapolation:
    258272
    259273\begin{equation}
    … …  
    261275\end{equation}
    262276
    263 Conditions for A, B and C constants are the same. Comparison of different LF extrapolation can be found on figure~\ref{lf}.
     277Conditions for the constants A, B and C  are the same. Comparison of different LF extrapolation can be found on figure~\ref{lf} and the performances of these methods on figure~\ref{lf2}.
    264278
    265279\begin{figure}[htbp]
    … …  
    282296
    283297
    284 In the rest of this paper was used the Gaussian method.
    285 
    286 
    287 Several high frequency (HF) extrapolation method were also tested several of them. The most common~\cite{VBthesis,DESYthesis} is :
     298In the rest of this paper we used the Gaussian method.
     299
     300
     301Several high frequency (HF) extrapolation method were also tested. The most common~\cite{VBthesis,DESYthesis} is :
    288302\begin{equation}
    289303\rho _ {HF} (\omega)=A\omega^{-4},
    … …  
    320334\end{itemize}
    321335
    322 These extrapolation methods are compared on figure~\ref{hf}.
     336These HF extrapolation methods are compared on figure~\ref{hf} and~\ref{hf2}.
    323337
    324338 
    … …  
    340354   \label{hf2}
    341355\end{figure}
     356
     357
     358
    342359\begin{figure}[htbp]
    343360 \centering
    … …  
    350367
    351368
    352 XXX If you show Lorenzian profiles for HF extrapolation why don't you also show it for LF extrapolation? XXX
     369\textbf{XXX If you show Lorenzian profiles for HF extrapolation why don't you also show it for LF extrapolation? XXX}
    353370
    354371
    … …  
    356373\section{Study of the reconstruction performance}
    357374
    358 After applying extrapolation and interpolation, the spectrum recovery is completed. Than was used  different reconstruction techniques to reconstruct the original profile. For each reconstruction method some profiles are very well reconstructed whereas some other are not so well reconstructed. Examples of well reconstructed profiles are shown on figure~\ref{good_profiles} and examples of poorly reconstructed profile are shown on figure~\ref{bad_profiles}.
     375After applying extrapolation and interpolation, the spectrum recovery is completed. Then we used  different reconstruction techniques to reconstruct the original profile. For each reconstruction method some profiles are very well reconstructed whereas some other are not so well reconstructed. Examples of well reconstructed profiles are shown on figure~\ref{good_profiles} and examples of poorly reconstructed profile are shown on figure~\ref{bad_profiles}.
    359376
    360377\begin{figure}[htbp]
    … …  
    379396
    380397
    381 The  $\Delta_{FWXM}$ and  $\chi^2$ distribution of the 1000 simulations which was maded and then reconstructed using the Hilbert transform method and Kramers-Kornig reconstruction are shown in on figure~\ref{profiles_stats_hilbert}. There is a good agreement in FWHM between the two methods indicating that they are both good at finding the bunch length. However, the Hilbert method gives lower $\chi^2$ indicating that this method is better at reconstruction the bunch profile.
     398The  $\Delta_{FWXM}$ and  $\chi^2$ distribution of the 1000 simulations which were made and then reconstructed using the Hilbert transform method and Kramers-Kornig reconstruction are shown in on figure~\ref{profiles_stats_hilbert}. There is a good agreement in FWHM between the two methods indicating that they are both good at finding the bunch length. However, the Hilbert method gives lower $\chi^2$ indicating that this method is better at reconstruction the bunch profile.
    382399
    383400
    … …  
    387404  \includegraphics*[width=70mm]{rev1/2.eps} \\
    388405  \includegraphics*[width=70mm]{new203/pic/3.eps}
    389   \caption{{$\Delta_{FWHM}$  (top)  and $\chi^2$ (bottom) distribution of 1000 simulations reconstructed using the Hilbert transform  method (black line) and Kramers-Kronig reconstruction method (red line).   XXX If the top figure is delta FWHM, then the title should say so XXX }}% VH change name of picture and unite with other
     406  \caption{{$\Delta_{FWHM}$  (top)  and $\chi^2$ (bottom) distribution of 1000 simulations reconstructed using the Hilbert transform  method (black line) and Kramers-Kronig reconstruction method (red line).  }}% VH change name of picture and unite with other
    390407   \label{profiles_stats_hilbert}
    391408\end{figure}
    392 \textbf{The fact that the phase recovery method based on the Kramers-Kronig relation gives a worst $\chi^2$  than the method based on Hilbert relation has been investigated. It is caused by the presence of negative components in the tails of the profiles. Figure~\ref{expKK} highlights this issue for one of the profiles.}
     409The fact that the phase recovery method based on the Kramers-Kronig relation gives a worst $\chi^2$  than the method based on Hilbert relation has been investigated. It is caused by the presence of negative components in the tails of the profiles. Figure~\ref{expKK} highlights this issue for one of the profiles.
    393410\begin{figure}[htbp]
    394411 \centering
    … …  
    421438
    422439
    423 While doing this work we also became aware of the discussion in~\cite{Pelliccia:2014vba} where it is argued that these reconstruction method have more difficulties with lorentzian profiles  than gaussian profiles. Therefore was simulated 1000 lorenzian profiles and performed a similar study. This is shown on figure~\ref{lorenz}. Although the $\chi^2$ is slightly worse in that case than in the case of gaussian profiles there still a good agreement between the original and reconstructed profiles.
     440While doing this work we also became aware of the discussion in~\cite{Pelliccia:2014vba} where it is argued that these reconstruction method have more difficulties with lorentzian profiles  than gaussian profiles. Therefore we simulated 1000 lorenzian profiles and performed a similar study. This is shown on figure~\ref{lorenz}. Although the $\chi^2$ is slightly worse in that case than in the case of gaussian profiles there still a good agreement between the original and reconstructed profiles.
    424441
    425442\begin{figure}[htbp]
    … …  
    434451
    435452% Effect of noise.
    436 So far was considered only the ideal case where no noise is added to the measured spectrum. However in a real experiment a noise component has to be added to the measured spectrum. This noise was added as follow :
     453In our discussion so far we considered only the ideal case where no noise is added to the measured spectrum. However in a real experiment a noise component has to be added to the measured spectrum. This noise was added as follow :
    437454\begin{equation}
    438455O_i' = O_i \times [1 + ( n_i N_{max}) ]
    439456\end{equation}
    440  where $O_i$ is the observed value,  $O_i'$ is the observed value with noise, $n_i$ is a random number between 0 and 1 (all numbers between 0 and 1 been equiprobable), and $N_{max}$ is the
     457 where $O_i$ is the observed value,  $O_i'$ is the observed value with noise, $n_i$ is a random number between 0 and 1 (all numbers between 0 and 1 being equiprobable), and $N_{max}$ is the
    441458maximum noise for that simulation (depending on the case this can be 5\%, 10\%, 20\%, 30\%, 40\% or  50\%). This study was done using linear sampling with 33 samples  and 1000 simulated profiles for each noise value. The figure~\ref{noise} shows how the $\chi^2$ is modified when this noise component is added.
    442459 
    … …  
    453470\section{Discussion}
    454471
    455 XXX To be updated XXX
    456 
    457 Was performed extensive simulation to estimate the performance of two phase recovery methods in the case of multi-gaussian and lorenzian profiles. In both cases was finded that when the sampling frequencies are chosen correctly  was obtained a good agreement between the original and reconstructed profiles (in most cases $\Delta_{FWXM} < 10\%$;  $\chi^2 ~ 10^{-6}$). This confirms that such methods are suitable to reconstruct the longitudinal profiles measured at particle accelerators using radiative methods.
    458 
    459 
     472We performed extensive simulation to estimate the performance of two phase recovery methods in the case of multi-gaussian and lorenzian profiles. In both cases we found that when the sampling frequencies are chosen correctly  we obtained a good agreement between the original and reconstructed profiles (in most cases $\Delta_{FWXM} < 10\%$;  $\chi^2 ~ 10^{-6}$). This confirms that such methods are suitable to reconstruct the longitudinal profiles measured at particle accelerators using radiative methods.
     473
     474
     475
     476The other are grateful for the funding received from the French ANR (contract ANR-12-JS05-0003-01), the PICS (CNRS) "Development of the instrumentation for accelerator experiments, beam monitoring and other applications" and  Research Grant \#F58/380-2013 (project F58/04) from the State Fund for Fundamental Researches of Ukraine in the frame of the State key laboratory of high energy physics.
     477
     478\clearpage
    460479\begin{thebibliography}{1}
    461480
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