Changeset 208 in ETALON for reconstruction


Ignore:
Timestamp:
Mar 18, 2015, 1:47:18 PM (9 years ago)
Author:
hodnevuc
Message:
 
Location:
reconstruction/long_paper3
Files:
81 added
6 edited

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  • reconstruction/long_paper3

  • reconstruction/long_paper3/phase_reconstruction_paper.aux

    r200 r208  
    44\citation{KK}
    55\citation{KK}
    6 \@LN@col{1}
    76\select@language{USenglish}
    87\@writefile{toc}{\select@language{USenglish}}
    … …  
    1110\@writefile{toc}{\contentsline {section}{Longitudinal bunch profile measurement at particle accelerators}{1}}
    1211\@writefile{toc}{\contentsline {section}{Reconstruction methods}{1}}
    13 \@LN@col{2}
    1412\@writefile{toc}{\contentsline {section}{Description of the simulations}{1}}
    1513\citation{E203prstab}
    16 \@LN@col{1}
    17 \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Example of profiles giving very different $\chi ^2$ despite being relatively similar.\relax }}{2}}
     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}}
    1815\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
    1916\newlabel{Offsine}{{1}{2}}
    20 \newlabel{eq:lamb}{{2}{2}}
    21 \@LN@col{2}
    22 \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Comparison of different samplings\relax }}{2}}
    23 \newlabel{samp}{{2}{2}}
     17\@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}}
     18\newlabel{sigma_chi2}{{2}{2}}
     19\newlabel{eq:lamb}{{7}{2}}
    2420\citation{pchip}
    2521\citation{VBthesis}
    26 \@LN@col{1}
    27 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.\relax }}{3}}
    28 \newlabel{lin12}{{3}{3}}
    29 \@LN@col{2}
    30 \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Comparison of different sampling with number of MDD. Ls -- is linear sampling with $1^o,5^o,10^0$ MDD and Triple sine sapmling; mx mean that in reconstruction was maximum number of detectors (blue and red on figure \ref  {lin12})\relax }}{3}}
    31 \newlabel{biglin}{{4}{3}}
    32 \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Effect of the sampling frequencies on the $\chi ^2$. \relax }}{3}}
    33 \newlabel{sampling_chi2}{{5}{3}}
     22\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Comparison of different samplings with $\chi ^2$ criterium (top) and $\Delta $ FWHM (bottom)\relax }}{3}}
     23\newlabel{samp}{{3}{3}}
     24\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.\relax }}{3}}
     25\newlabel{lin12}{{4}{3}}
    3426\citation{VBthesis,DESYthesis}
    3527\citation{LaiS}
    36 \@LN@col{1}
    37 \@LN@col{2}
    38 \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Comparison of different LF extrapolation: example of spectrum (top), profile (middle) and histogram with mean $\chi ^2$ for each method (bottom). Gauss and Taylor methods are described in the text. "Real LF spectrum" means that the real LF spectrum is used. For this simulation we use the Hilbert method of phase recovery and $A\omega ^B$ high frequency extrapolation. \relax }}{4}}
    39 \newlabel{lf}{{6}{4}}
    40 \@LN@col{1}
    41 \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Comparison of different HF extrapolation~: example of spectrum (top) and profile (upper middle), histogram with mean $\chi ^2$ for comparison for Gaussian profiles (lower middle) and Lorenzians (bottom). For these simulation we use the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations.\relax }}{5}}
    42 \newlabel{hf}{{7}{5}}
     28\@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 Triple sine sapmling; mx mean that in reconstruction was maximum number of detectors (blue and red on figure \ref  {lin12})\relax }}{4}}
     29\newlabel{biglin}{{5}{4}}
     30\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Effect of the sampling frequencies on the $\chi ^2$ (top) and $\Delta $ FWHM (bottom). \relax }}{4}}
     31\newlabel{sampling_chi2}{{6}{4}}
     32\citation{Pelliccia:2014vba}
     33\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Comparison of different LF extrapolation: example of spectrum (top) and profile (bottom) and histogram with mean $\chi ^2$ for each method (bottom). Gaussian and Taylorian methods are described in the text. "Real LF spectrum" means that the real LF spectrum is used. For this simulation was used the Hilbert method of phase recovery and $A\omega ^B$ high frequency extrapolation.\relax }}{5}}
     34\newlabel{lf}{{7}{5}}
    4335\@writefile{toc}{\contentsline {section}{Study of the reconstruction performance}{5}}
    44 \@LN@col{2}
    45 \@writefile{lof}{\contentsline {figure}{\numberline {8}{\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 }}{5}}
    46 \newlabel{good_profiles}{{8}{5}}
    47 \citation{Pelliccia:2014vba}
    48 \@LN@col{1}
    49 \@writefile{lof}{\contentsline {figure}{\numberline {9}{\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 }}{6}}
    50 \newlabel{bad_profiles}{{9}{6}}
    51 \@LN@col{2}
    52 \@writefile{lof}{\contentsline {figure}{\numberline {10}{\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 }}{6}}
    53 \newlabel{profiles_stats_hilbert}{{10}{6}}
    54 \@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces $\Delta _{FWXM}$ for 1000 profiles with both methods.\relax }}{6}}
    55 \newlabel{fwxm}{{11}{6}}
    56 \@LN@col{1}
    57 \@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Original and reconstructed profile and their difference for two different profiles.\relax }}{7}}
    58 \newlabel{mod}{{12}{7}}
    59 \@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Effect of scaling the constraints on the parameters $\sigma _i$ and $\mu _i$ on the $\chi ^2$.\relax }}{7}}
    60 \newlabel{sigma_chi2}{{13}{7}}
    61 \@LN@col{2}
    62 \@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces Distribution of the $\chi ^2$ in the case of a lorenzian distribution.\relax }}{7}}
    63 \newlabel{lorenz}{{14}{7}}
    64 \@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Mean $\chi ^2$ as function of noise amplitude.\relax }}{7}}
    65 \newlabel{noise}{{15}{7}}
    66 \@writefile{toc}{\contentsline {section}{Discussion}{7}}
     36\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Comparison of different LF extrapolation: histogram with mean $\chi ^2$ for each method (top) and $\Delta $ FWHM (bottom). \relax }}{5}}
     37\newlabel{lf2}{{8}{5}}
    6738\bibcite{OTR_LURE}{1}
    6839\bibcite{ODR_Cianchi}{2}
    6940\bibcite{Doucas_ESB}{3}
    7041\bibcite{KK}{4}
     42\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Comparison of different HF extrapolation~: example of spectrum (top) and profile (bottom). For these simulation was used the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations.\relax }}{6}}
     43\newlabel{hf}{{9}{6}}
     44\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Comparison of different HF extrapolation for Gaussian~:histogram with mean $\chi ^2$ (top) and $\Delta $ FWHM (bottom).\relax }}{6}}
     45\newlabel{hf2}{{10}{6}}
     46\@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}}
     47\newlabel{hf3}{{11}{6}}
     48\@writefile{toc}{\contentsline {section}{Discussion}{6}}
    7149\bibcite{E203prstab}{5}
    7250\bibcite{Pelliccia:2014vba}{6}
    … …  
    7553\bibcite{LaiS}{9}
    7654\bibcite{DESYthesis}{10}
    77 \@LN@col{1}
    78 \@LN@col{2}
     55\@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}}
     56\newlabel{good_profiles}{{12}{7}}
     57\@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}}
     58\newlabel{bad_profiles}{{13}{7}}
     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}}
     60\newlabel{profiles_stats_hilbert}{{14}{8}}
     61\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Explanation of $\chi ^2$ distribution\relax }}{8}}
     62\newlabel{expKK}{{15}{8}}
     63\@writefile{lof}{\contentsline {figure}{\numberline {16}{\ignorespaces $\Delta _{FWXM}$ for 1000 profiles with both methods.\relax }}{8}}
     64\newlabel{fwxm}{{16}{8}}
     65\@writefile{lof}{\contentsline {figure}{\numberline {17}{\ignorespaces Original and reconstructed profile and their difference for bad profile (top) and good profile (bottom).\relax }}{8}}
     66\newlabel{mod}{{17}{8}}
     67\@writefile{lof}{\contentsline {figure}{\numberline {18}{\ignorespaces Distribution of the $\chi ^2$ in the case of a lorenzian distribution. XXX Can you add delta FWHM? XXX\relax }}{8}}
     68\newlabel{lorenz}{{18}{8}}
     69\@writefile{lof}{\contentsline {figure}{\numberline {19}{\ignorespaces Mean $\chi ^2$ as function of noise amplitude. XXX Can you add delta FWHM? XXX\relax }}{9}}
     70\newlabel{noise}{{19}{9}}

  • reconstruction/long_paper3/phase_reconstruction_paper.log

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     1820(epstopdf)             Command: <repstopdf --outfile=rev4/hflor-eps-converted-t
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     1834(epstopdf)                    size: 15268 bytes
     1835(epstopdf)             Output file: <rev1/541-eps-converted-to.pdf>
     1836(epstopdf)                    date: 2015-03-17 16:08:27
     1837(epstopdf)                    size: 11062 bytes
     1838(epstopdf)             Command: <repstopdf --outfile=rev1/541-eps-converted-to.
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    1728 (epstopdf)             Output file: <new203/pic/658-eps-converted-to.pdf>
    1729 (epstopdf)                    date: 2015-03-05 13:30:00
    1730 (epstopdf)                    size: 10393 bytes
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     1852(epstopdf)             Output file: <rev1/658-eps-converted-to.pdf>
     1853(epstopdf)                    date: 2015-03-17 16:08:27
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    1746 (epstopdf)             Output file: <new203/pic/914-eps-converted-to.pdf>
    1747 (epstopdf)                    date: 2015-03-05 13:30:00
    1748 (epstopdf)                    size: 10930 bytes
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     1869(epstopdf)             Output file: <rev1/914-eps-converted-to.pdf>
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     1871(epstopdf)                    size: 11137 bytes
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    1764 (epstopdf)             Output file: <new203/pic/227-eps-converted-to.pdf>
    1765 (epstopdf)                    date: 2015-03-05 13:30:00
    1766 (epstopdf)                    size: 11734 bytes
    1767 (epstopdf)             Command: <repstopdf --outfile=new203/pic/227-eps-convert
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     1886(epstopdf)             Output file: <rev1/227-eps-converted-to.pdf>
     1887(epstopdf)                    date: 2015-03-17 16:08:28
     1888(epstopdf)                    size: 12120 bytes
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    1782 (epstopdf)             Output file: <new203/pic/231-eps-converted-to.pdf>
    1783 (epstopdf)                    date: 2015-03-05 13:30:00
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    1785 (epstopdf)             Command: <repstopdf --outfile=new203/pic/231-eps-convert
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     1903(epstopdf)             Output file: <rev1/231-eps-converted-to.pdf>
     1904(epstopdf)                    date: 2015-03-17 16:08:29
     1905(epstopdf)                    size: 11088 bytes
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     1936(epstopdf)                    size: 12227 bytes
     1937(epstopdf)             Output file: <rev1/2-eps-converted-to.pdf>
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     1939(epstopdf)                    size: 8493 bytes
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    1837 (epstopdf)                    date: 2015-03-05 13:30:00
     1955(epstopdf)                    date: 2015-03-02 10:34:35
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    1852 Underfull \vbox (badness 10000) has occurred while \output is active []
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  • reconstruction/long_paper3/phase_reconstruction_paper.tex

    r200 r208  
    44               %refpage      % separate references
    55              ]{jacow}
    6 \usepackage{lineno}
     6%\usepackage{lineno}
    77
    88\makeatletter%                           % test for XeTeX where the sequence is by default eps-> pdf, jpg, png, pdf, ...
    … …  
    4242
    4343\begin{document}
    44 \linenumbers
     44%\linenumbers
    4545\title{Study of Phase Reconstruction Techniques applied to Smith-Purcell Radiation Measurements\thanks{Work supported by 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." }}
    4646
    … …  
    6767where $I(\lambda)$ 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.
    6868
    69 A phase reconstruction algorithm must therefore be used to recover this phase. Several methods exist (see for example~\cite{KK}). We have implemented two of these methods and we assess their performances below.
     69A 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.
    7070
    7171\section{Reconstruction methods}
    … …  
    8484To recover the phase from the amplitude, the function should  be written as: $log(F(\omega))=log(\rho(\omega))+i\Theta(\omega)$ with $\rho(\omega)$ its amplitude and $\Theta(\omega)$ its phase.
    8585The Kramers-Kronig relations can then be applied as follows:
    86 $$\Theta(\omega_0)  =  \frac{2\omega_0}{\pi} \textit{P}\int^{+ \infty}_{0}\frac{ln(\rho(\omega) )}{\omega_0^2-\omega^2}d\omega$$
     86\begin{equation}
     87\Theta(\omega_0)  =  \frac{2\omega_0}{\pi} \textit{P}\int^{+ \infty}_{0}\frac{ln(\rho(\omega) )}{\omega_0^2-\omega^2}d\omega
     88\end{equation}
    8789The basis of this relationship are the Cauchy-Riemann conditions (analyticity of function).  %In this case, the value spectrum can gain value at [0, $\infty$).\par
    8890
    8991
    9092In some cases this phase can also be obtained simply by using the Hilbert transform of the spectrum:
    91 $$\Theta(\omega_0)  =  -\frac{1}{\pi} \textit{P}\int^{+ \infty}_{- \infty}\frac{ln(\rho(\omega))}{\omega_0-\omega}d\omega.$$
    92 {As the Hilbert transform ($\textit{H}$) is related to the Fourier transform (${\cal F}$): $${\cal F}(\textit{H}(u))(\omega)=(-isgn(\omega)){\cal F}(u)(\omega),$$ the calculation of phase can use optimised FFT code and is much faster than  calculating the Kramers-Kronig's integral.}%VH add text
    93 We have 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.
     93\begin{equation}
     94\Theta(\omega_0)  =  -\frac{1}{\pi} \textit{P}\int^{+ \infty}_{- \infty}\frac{ln(\rho(\omega))}{\omega_0-\omega}d\omega.
     95\end{equation}
     96{As the Hilbert transform ($\textit{H}$) is related to the Fourier transform (${\cal F}$):
     97\begin{equation}
     98{\cal F}(\textit{H}(u))(\omega)=(-isgn(\omega)){\cal F}(u)(\omega),
     99\end{equation}
     100the calculation of phase can use optimised FFT code and is much faster than  calculating the Kramers-Kronig's integral.}%VH add text
     101Was 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.
    94102
    95103\section{Description of the simulations}
    96104
    97 To test the performance of these methods we have 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.
     105To 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.
    98106% 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).
    99 We have checked that our conclusions are valid across this range. }
     107Was checked that our conclusions are valid across this range. }
    100108% VH add and change text
    101109
    102110
    103 Using this formula we have generated 1000 profiles, we 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).
    104 
    105 To estimate the performance of the reconstruction several estimators are available. We choose to use the $\chi^2$, defined as follow:
    106 $$\chi^2=\sum_i\omega_i^2(O_i-E_i)^2/N,$$
     111Using 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
     113To estimate the performance of the reconstruction several estimators are available. Was choosed to use the $\chi^2$, defined as follow:
     114\begin{equation}
     115\chi^2=\sum_i\omega_i^2(O_i-E_i)^2/N,
     116\end{equation}
    107117where $O_i$ 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
    108118However 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.
    109119
    110 \begin{figure}[!htb]
    111  \centering
    112   \includegraphics*[width=70mm]{a.eps}
    113   \caption{Example of profiles giving very different $\chi^2$ despite being relatively similar.}% VH change name of picture and unite with other
    114    \label{Offsine}
    115 \end{figure}
    116  So we decided to also look at the FWHM which we generalized as FWXM where $X \in [0.1 ; 0.9]$ is the fraction of the maximum value at which we calculate 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 (as needed). We have created an estimator $\Delta_{FWXM}$ defined as follow:
     120
     121Also 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:
    117122
    118123%$$
    119124%\Delta_{FWXM} = \mbox{Max}_{X \in \mbox{rset} }\left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right|
    120125%$$
    121 $$
     126\begin{equation}
    122127\Delta_{FWXM} = \left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right|
    123 $$
     128\end{equation}
    124129where $\mbox{rset} = \{ 0.1 ; 0.2 ; 0.5 ; 0.8 ; 0.9\}$, $FWXM_{\mbox{orig}}$ and $FWXM_{\mbox{reco}}$ are the FWXM of the original and reconstructed profiles respectively.
    125130
    126 
     131\begin{figure}[!htb]
     132 \centering
     133  \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
     136   \label{Offsine}
     137\end{figure}
    127138To 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} XXX can you add FWHM ? XXX.
    128139
    … …  
    131142  \includegraphics*[width=70mm]{newfigures/chi_sigma.eps}\\
    132143  \includegraphics*[width=70mm]{newfigures/chi_mu.eps}%add new figure
    133   \caption{Effect of scaling the constraints on the parameters $\sigma_i$ and $\mu_i$ on the $\chi^2$.}
     144  \caption{Effect of scaling the constraints on the parameters $\sigma_i$ (top) and $\mu_i$ (bottom) on the $\chi^2$.}
    134145   \label{sigma_chi2}
    135146\end{figure}
    … …  
    147158with $l_n =50, 250, 1500 \mu m$ and $\Theta$ varying between $40^o$ and $140^o$.
    148159Uniform location of detectors in space corresponds to the inhomogeneous sample frequency and vice versa. %So next sampling is linear in frequecy.
    149 \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, we use formula:
    150 $$\omega_0+(\omega_f-\omega_0)/32\times(0:32).$$
    151 \item \textit{Logarithmic sampling}. Point is distributed according logarithmic low. For this, we use next formula:
    152 $$\omega_0*exp(log(\omega_f/\omega_0)\times(0:32)/32).$$
     160\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:
     161\begin{equation}
     162\omega_i=\omega_0+(\omega_f-\omega_0)/32\times(0:32).
     163\end{equation}
     164\item \textit{Logarithmic sampling}. Point is distributed according logarithmic low. For this, was used:
     165\begin{equation}
     166\omega_0*exp(log(\omega_f/\omega_0)\times(0:32)/32).
     167\end{equation}
    153168For this sampling first and last points is the same as in Triple-sine sampling. This was done to avoid impact of extrapolations on result.
    154169%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.
    155170\end{itemize}
    156 \textbf{The study of the sampling is important, as it tells how to best position the detectors and to optimize system. Linear sample spectrum gives the best result (see figure (\ref{samp})).
    157 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. 
     171{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 (see figure (\ref{samp})).
     172%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. 
    158173}
    159174
    160175\begin{figure}[!htb]
    161176 \centering
    162   \includegraphics*[width=70mm]{new203/pic/4.eps} \\
    163     \includegraphics*[width=70mm]{new203/pic/5.eps}
     177  \includegraphics*[width=70mm]{rev1/4.eps} \\
     178    \includegraphics*[width=70mm]{rev1/5.eps}
    164179  %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps}
    165   \caption{Comparison of different samplings}%VH add  picture
     180  \caption{Comparison of different samplings with $\chi^2$ criterium (top) and $\Delta$ FWHM (bottom)}%VH add  picture
    166181   \label{samp}
    167182\end{figure}
    168 \textbf{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.
    169 For angle calculation and applying condition for first and final point we use  formula~\eqref{eq:lamb}.
    170 On figure~\ref{lin12} examples of detector positions are shown. The position of the red points is calculated by formula~\ref{XXX} and the blue are the possible positions of detector which does not break minimum detector distance (MDD) XXX This is incomplete XXX.}
    171 \begin{figure}[!htb]
    172  \centering
    173   \includegraphics*[width=70mm]{new203/lin1.eps} \\
    174     \includegraphics*[width=70mm]{new203/lin2.eps}
     183{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.
     184%For angle calculation and applying condition for first and final point was used  formula ~\ref{eq:lamb}.
     185On 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.}
     186\begin{figure}[!htb]
     187 \centering
     188  \includegraphics*[width=70mm]{rev1/mmd5.eps} \\
     189    \includegraphics*[width=70mm]{rev1/mmd10.eps}
    175190  %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps}
    176191  \caption{Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.}%VH add  picture
    … …  
    179194
    180195\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).
    181 As the Lsmx configuration is physically impossible, 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.}\par
    182 \begin{figure}[!htb]
    183  \centering
    184   \includegraphics*[width=90mm]{new203/histLINEAR.eps}
    185  
    186   \caption{Comparison of different sampling with number of MDD. Ls -- is linear sampling with $1^o,5^o,10^0$ MDD and Triple sine sapmling; mx mean that in reconstruction was maximum number of detectors (blue and red on figure \ref{lin12})}%VH add  picture
     196%As the Lsmx configuration is physically impossible,
     197So 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.}\par
     198\begin{figure}[!htb]
     199 \centering
     200  \includegraphics*[width=90mm]{rev1/hist1.eps} \\
     201  \includegraphics*[width=90mm]{rev1/hist1fw.eps}
     202  \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 Triple sine sapmling; mx mean that in reconstruction was maximum number of detectors (blue and red on figure \ref{lin12})}%VH add  picture
    187203   \label{biglin}
    188204\end{figure}
    … …  
    190206
    191207%This section has been moved from elsewhere
    192 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 we used the same spectrum but sampled  with 5 to 120  points. The effect of changing the sampling frequencies on the  $\chi^2$ is shown on figure~\ref{sampling_chi2}. This study uses linear sampling with 1000 profiles for each point and the Hilbert reconstruction method.
    193 
    194 XXX Why you did not do the figure for FWHM ? XXX
    195 
    196 \begin{figure}[!htb]
    197  \centering
    198   \includegraphics*[width=70mm]{newfigures/Chi_Ndet.eps}
    199   \caption{Effect of the sampling frequencies on the $\chi^2$. }
     208The 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.
     209
     210
     211
     212\begin{figure}[!htb]
     213 \centering
     214  \includegraphics*[width=70mm]{rev4/chN.eps}\\
     215   \includegraphics*[width=70mm]{rev4/fwN.eps}
     216  \caption{Effect of the sampling frequencies on the $\chi^2$ (top) and $\Delta$ FWHM (bottom). }
    200217   \label{sampling_chi2}
    201218\end{figure}
    202 
     219It can be seen at figure \ref{sampling_chi2} that beyond 33 sampling points the gain on the reconstructed $\chi^2$ is marginal.
    203220
    204221
    … …  
    206223
    207224After 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}.
    208 The interpolation function must satisfy the following criteria: it must conserve the slope at the two endpoints (to have a continuous derivative) and respects monotonicity. Cubic Hermite interpolation has been chosen as it matches these requirements. XXX Something is not clear here: which function was used for interpolation? PCHIP or Cubic hermite ? XXX
     225The 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.
    209226
    210227For low frequency extrapolation two methods have been investigated: Gaussian or Taylorian.
    211228
    212 In the Gaussian method, we define the extrapolation as follow:
     229In the Gaussian method, was defined the extrapolation as follow:
    213230\begin{equation}
    214231\rho_{LF}(\omega)=Ae^{-(\omega-B)^2/2C^2}
    … …  
    233250Approximation to the 4th order gives the following LF extrapolation:
    234251
    235 $$\rho_{LF}=|F(\omega)|=\sqrt{A+B\omega^2+C\omega^4}$$
     252\begin{equation}
     253\rho_{LF}=|F(\omega)|=\sqrt{A+B\omega^2+C\omega^4}
     254\end{equation}
    236255
    237256Conditions for A, B and C constants are the same. Comparison of different LF extrapolation can be found on figure~\ref{lf}.
    … …  
    239258\begin{figure}[!htb]
    240259 \centering
    241   \includegraphics*[width=65mm]{new203/LFsp.eps}\\
    242   \includegraphics*[width=65mm]{new203/LFpr.eps}\\
    243     \includegraphics*[width=65mm]{new203/LF.eps}
    244     \caption{Comparison of different LF extrapolation: example of spectrum (top), profile (middle) and histogram with mean $\chi^2$ for each method (bottom). Gauss and Taylor methods are described in the text. "Real LF spectrum" means that the real LF spectrum is used. For this simulation we use the Hilbert method of phase recovery and $A\omega^B$ high frequency extrapolation.     }
     260  \includegraphics*[width=65mm]{rev2/lfsp.eps}\\
     261  \includegraphics*[width=65mm]{rev2/lfpr.eps}\\
     262
     263    \caption{Comparison of different LF extrapolation: example of spectrum (top) and  profile (bottom) and histogram with mean $\chi^2$ for each method (bottom). Gaussian and Taylorian methods are described in the text. "Real LF spectrum" means that the real LF spectrum is used. For this simulation was used the Hilbert method of phase recovery and $A\omega^B$ high frequency extrapolation.}
    245264   \label{lf}
    246265\end{figure}
    247266
    248 In the rest of this paper we use the Gaussian method.
     267\begin{figure}[!htb]
     268 \centering
     269
     270  \includegraphics*[width=65mm]{new203/LF.eps}
     271    \includegraphics*[width=65mm]{rev3/lffw.eps}
     272    \caption{Comparison of different LF extrapolation: histogram with mean $\chi^2$ for each method (top) and $\Delta$ FWHM (bottom).  }
     273   \label{lf2}
     274\end{figure}
     275
     276
     277In the rest of this paper was used the Gaussian method.
    249278
    250279
    251280Several high frequency (HF) extrapolation method were also tested several of them. The most common~\cite{VBthesis,DESYthesis} is :
    252 $$\rho _ {HF} (\omega)=A\omega^{-4},$$ where $\rho_ {HF} (\omega)$ is the extrapolated spectrum at high frequency and $A=\rho_H \omega_H^{4} $. XXX H is not defined here XXX
     281\begin{equation}
     282\rho _ {HF} (\omega)=A\omega^{-4},
     283\end{equation}
     284 where $\rho_ {HF} (\omega)$ is the extrapolated spectrum at high frequency and $A=\rho_f \omega_f^{4} $, where $\rho_f$ is spectrum value of final point $\omega_f$.
    253285
    254286The second method use the same consideration as in Lai and Sievers~\cite{LaiS}:
    … …  
    267299\end{itemize}
    268300where $\omega_{fmax}$  is the last sampled point  of the spectrum. To satisfy the boundary condition two constants are needed, giving a two-terms extrapolation~:
    269 $$\rho_{HF}(\omega)=A\omega^{-2}+B\omega^{-3}$$
     301\begin{equation}
     302\rho_{HF}(\omega)=A\omega^{-2}+B\omega^{-3}
     303\end{equation}
    270304or  extrapolation with the exponent as free parameter:
    271 $$\rho_{HF}(\omega)=A\omega^B$$
     305\begin{equation}
     306\rho_{HF}(\omega)=A\omega^B
     307\end{equation}
    272308where the A and B coefficient are selected from the boundary conditions.
    273309Two other extrapolation methods have also been investigated:
    … …  
    283319\begin{figure}[!htb]
    284320 \centering
    285   \includegraphics*[width=65mm]{new203/HFsp.eps}\\
    286   \includegraphics*[width=65mm]{new203/HFprofile.eps}\\
    287     \includegraphics*[width=65mm]{new203/HFGauss.eps}
    288         \includegraphics*[width=65mm]{new203/HFLorenz.eps}
    289     \caption{Comparison of different HF extrapolation~: example of spectrum  (top) and profile (upper middle), histogram with mean $\chi^2$ for comparison for Gaussian profiles (lower middle) and Lorenzians (bottom). For these simulation we use the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations.}
     321  \includegraphics*[width=65mm]{rev2/hfsp.eps}\\
     322  \includegraphics*[width=65mm]{rev2/hfpr.eps}\\
     323
     324    \caption{Comparison of different HF extrapolation~: example of spectrum  (top) and profile (bottom). For these simulation was used the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations.}
    290325   \label{hf}
    291326\end{figure}
     327\begin{figure}[!htb]
     328 \centering
     329    \includegraphics*[width=65mm]{new203/HFGauss.eps}\\
     330        \includegraphics*[width=65mm]{rev4/hfgas.eps}\\
     331
     332    \caption{Comparison of different HF extrapolation for Gaussian~:histogram with mean $\chi^2$ (top) and $\Delta$ FWHM (bottom).}
     333   \label{hf2}
     334\end{figure}
     335\begin{figure}[!htb]
     336 \centering
     337        \includegraphics*[width=65mm]{new203/HFLorenz.eps}\\
     338          \includegraphics*[width=65mm]{rev4/hflor.eps}
     339        \caption{Comparison of different HF extrapolation for Lorenzians~:histogram with mean $\chi^2$ (top) and $\Delta$ FWHM (bottom).}
     340   \label{hf3}
     341\end{figure}
     342
     343
    292344
    293345XXX If you show Lorenzian profiles for HF extrapolation why don't you also show it for LF extrapolation? XXX
    … …  
    297349\section{Study of the reconstruction performance}
    298350
    299 After applying extrapolation and interpolation the spectrum recovery is complete and we can apply 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}.
     351After 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}.
    300352
    301353\begin{figure}[!htb]
    302354 \centering
    303355%  \includegraphics*[trim=0 0 275 0 ,clip,width=95mm]{plot1000700.png}
    304   \includegraphics*[width=65mm]{new203/pic/541.eps}\\
    305   \includegraphics*[width=65mm]{new203/pic/658.eps}\\
    306    \includegraphics*[width=65mm]{new203/pic/914.eps}
     356  \includegraphics*[width=65mm]{rev1/541.eps}\\
     357  \includegraphics*[width=65mm]{rev1/658.eps}\\
     358   \includegraphics*[width=65mm]{rev1/914.eps}
    307359  \caption{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.}
    308360   \label{good_profiles}
    … …  
    312364 \centering
    313365%  \includegraphics*[trim=0 0 275 0 ,clip,width=95mm]{plot1000183.eps}
    314   \includegraphics*[width=65mm]{new203/pic/227.eps}\\
    315    \includegraphics*[width=65mm]{new203/pic/231.eps}\\
    316     \includegraphics*[width=65mm]{new203/pic/667.eps}\\
     366  \includegraphics*[width=65mm]{rev1/227.eps}\\
     367   \includegraphics*[width=65mm]{rev1/231.eps}\\
     368    \includegraphics*[width=65mm]{rev1/667.eps}\\
    317369  \caption{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.}
    318370   \label{bad_profiles}
    … …  
    320372
    321373
    322 The  $\Delta_{FWXM}$ and  $\chi^2$ distribution of the 1000 simulations which we 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 we see that the Hilbert method gives lower $\chi^2$ indicating that this method is better at reconstruction the bunch profile.
    323 
    324 
    325 
    326 \begin{figure}[!htb]
    327  \centering
    328   \includegraphics*[width=70mm]{new203/pic/2.eps} \\
     374The  $\Delta_{FWXM}$ 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.
     375
     376
     377
     378\begin{figure}[!htb]
     379 \centering
     380  \includegraphics*[width=70mm]{rev1/2.eps} \\
    329381  \includegraphics*[width=70mm]{new203/pic/3.eps}
    330382  \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
    331383   \label{profiles_stats_hilbert}
    332384\end{figure}
    333 
    334 
     385\textbf{The fact that phase recovery based on Kramers-Kronig relation work worst than Hilbert method is caused by negative part of tails of profiles. At figure \ref{expKK} is shown example of one of the profiles.}
     386\begin{figure}[!htb]
     387 \centering
     388  \includegraphics*[width=70mm]{rev3/compFig.eps} 
     389  \caption{Explanation of $\chi^2$ distribution}% VH change name of picture and unite with other
     390   \label{expKK}
     391\end{figure}
    335392% VH add this block
    336393%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    … …  
    338395 Figure~\ref{mod} shows the modulus of the difference between the original and reconstructed profiles. One can see oscillations in the difference between the original and reconstructed profile.
    339396 
    340      XXX Figure \ref{mod}  needs to be fixed:
    341      Please use subplot to split the figure in 2 parts:
    342      The upper 66\% of the figure should show the original and reconstructed profile.
    343      The lower 33\% should show the difference (your dashed line). At the moment we have no indication of the scale to be applied to the dashed line. It would be important to have one.
    344      XXX
     397   
    345398     
    346399\begin{figure}[!htb]
    347400 \centering
    348   \includegraphics*[width=70mm]{new203/pic/1.eps}
     401  \includegraphics*[width=70mm]{rev1/1.eps}
    349402  \caption{$\Delta_{FWXM}$  for 1000 profiles with both methods.}%VH add  picture
    350403   \label{fwxm}
    … …  
    352405\begin{figure}[!htb]
    353406 \centering
    354   \includegraphics*[width=65mm]{new203/pic/6.eps}\\
    355     \includegraphics*[width=65mm]{new203/pic/7.eps}
    356     \caption{Original and reconstructed profile and their difference for two different profiles.}
     407  \includegraphics*[width=65mm]{rev4/d1.eps}\\
     408    \includegraphics*[width=65mm]{rev4/d2.eps}
     409    \caption{Original and reconstructed profile and their difference for bad profile (top) and good profile (bottom).}
    357410   \label{mod}
    358411\end{figure}
    … …  
    360413
    361414
    362 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 we also 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 we still have a good agreement between the original and reconstructed profiles.
     415While 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.
    363416
    364417\begin{figure}[!htb]
    … …  
    372425
    373426% Effect of noise.
    374 So far we have only considered 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 :
    375 $$O_i' = O_i \times (1 + n_i) N_{max}$$
    376 
    377 XXX I think it should be XXX
    378 $$O_i' = O_i \times [1 + ( n_i N_{max}) ] $$
    379  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 XXX Please check if this is true or if your random function has a gaussian distribution XXX), and $N_{max}$ is the
     427So 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 :
     428\begin{equation}
     429O_i' = O_i \times [1 + ( n_i N_{max}) ]
     430\end{equation}
     431 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
    380432maximum 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.
    381433 
    … …  
    393445XXX To be updated XXX
    394446
    395 We have 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 find that when the sampling frequencies are chosen correctly  we obtain 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.
     447Was 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.
    396448
    397449
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