Changeset 208 in ETALON for reconstruction/long_paper3/phase_reconstruction_paper.tex

Timestamp:

Mar 18, 2015, 1:47:18 PM (10 years ago)

Author:

hodnevuc

Message:

 

Location:

reconstruction/long_paper3

Files:

• . (modified) (1 prop)
• phase_reconstruction_paper.tex (modified) (21 diffs)

reconstruction/long_paper3/phase_reconstruction_paper.tex

@@ -4 +4 @@
                %refpage      % separate references
               ]{jacow}
-\usepackage{lineno}
+%\usepackage{lineno}
 
 \makeatletter%                           % test for XeTeX where the sequence is by default eps-> pdf, jpg, png, pdf, ...
@@ -42 +42 @@
 
 \begin{document}
-\linenumbers
+%\linenumbers
 \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." }}
 
@@ -67 +67 @@
 where $I(\lambda)$ 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.
 
-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.
+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.
 
 \section{Reconstruction methods}
@@ -84 +84 @@
 To 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. 
 The Kramers-Kronig relations can then be applied as follows:
-$$\Theta(\omega_0)  =  \frac{2\omega_0}{\pi} \textit{P}\int^{+ \infty}_{0}\frac{ln(\rho(\omega) )}{\omega_0^2-\omega^2}d\omega$$
+\begin{equation}
+\Theta(\omega_0)  =  \frac{2\omega_0}{\pi} \textit{P}\int^{+ \infty}_{0}\frac{ln(\rho(\omega) )}{\omega_0^2-\omega^2}d\omega
+\end{equation}
 The 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
 
 
 In some cases this phase can also be obtained simply by using the Hilbert transform of the spectrum:
-$$\Theta(\omega_0)  =  -\frac{1}{\pi} \textit{P}\int^{+ \infty}_{- \infty}\frac{ln(\rho(\omega))}{\omega_0-\omega}d\omega.$$
-{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
-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.
+\begin{equation}
+\Theta(\omega_0)  =  -\frac{1}{\pi} \textit{P}\int^{+ \infty}_{- \infty}\frac{ln(\rho(\omega))}{\omega_0-\omega}d\omega.
+\end{equation}
+{As the Hilbert transform ($\textit{H}$) is related to the Fourier transform (${\cal F}$): 
+\begin{equation}
+{\cal F}(\textit{H}(u))(\omega)=(-isgn(\omega)){\cal F}(u)(\omega),
+\end{equation}
+the calculation of phase can use optimised FFT code and is much faster than  calculating the Kramers-Kronig's integral.}%VH add text
+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.
 
 \section{Description of the simulations}
 
-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.
+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.
 % 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). 
-We have checked that our conclusions are valid across this range. }
+Was checked that our conclusions are valid across this range. }
 % VH add and change text
 
 
-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).
-
-To estimate the performance of the reconstruction several estimators are available. We choose to use the $\chi^2$, defined as follow:
-$$\chi^2=\sum_i\omega_i^2(O_i-E_i)^2/N,$$
+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).
+
+To estimate the performance of the reconstruction several estimators are available. Was choosed to use the $\chi^2$, defined as follow:
+\begin{equation}
+\chi^2=\sum_i\omega_i^2(O_i-E_i)^2/N,
+\end{equation}
 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
 However 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.
 
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=70mm]{a.eps}
-  \caption{Example of profiles giving very different $\chi^2$ despite being relatively similar.}% VH change name of picture and unite with other
-   \label{Offsine}
-\end{figure}
- 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:
+
+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:
 
 %$$
 %\Delta_{FWXM} = \mbox{Max}_{X \in \mbox{rset} }\left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right| 
 %$$
-$$
+\begin{equation}
 \Delta_{FWXM} = \left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right| 
-$$
+\end{equation}
 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.
 
-
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{rev1/chiexp.eps}
+  \caption{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. }% VH change name of picture and unite with other
+   \label{Offsine}
+\end{figure}
 To 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.
 
@@ -131 +142 @@
   \includegraphics*[width=70mm]{newfigures/chi_sigma.eps}\\
   \includegraphics*[width=70mm]{newfigures/chi_mu.eps}%add new figure
-  \caption{Effect of scaling the constraints on the parameters $\sigma_i$ and $\mu_i$ on the $\chi^2$.}
+  \caption{Effect of scaling the constraints on the parameters $\sigma_i$ (top) and $\mu_i$ (bottom) on the $\chi^2$.}
    \label{sigma_chi2}
 \end{figure}
@@ -147 +158 @@
 with $l_n =50, 250, 1500 \mu m$ and $\Theta$ varying between $40^o$ and $140^o$.
 Uniform location of detectors in space corresponds to the inhomogeneous sample frequency and vice versa. %So next sampling is linear in frequecy.
-\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:
-$$\omega_0+(\omega_f-\omega_0)/32\times(0:32).$$
-\item \textit{Logarithmic sampling}. Point is distributed according logarithmic low. For this, we use next formula:
-$$\omega_0*exp(log(\omega_f/\omega_0)\times(0:32)/32).$$
+\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:
+\begin{equation}
+\omega_i=\omega_0+(\omega_f-\omega_0)/32\times(0:32).
+\end{equation}
+\item \textit{Logarithmic sampling}. Point is distributed according logarithmic low. For this, was used:
+\begin{equation}
+\omega_0*exp(log(\omega_f/\omega_0)\times(0:32)/32).
+\end{equation}
 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. 
 %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.
 \end{itemize}
-\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})).
-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.  
+{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})).
+%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.  
 }
 
 \begin{figure}[!htb]
  \centering
-  \includegraphics*[width=70mm]{new203/pic/4.eps} \\
-    \includegraphics*[width=70mm]{new203/pic/5.eps}
+  \includegraphics*[width=70mm]{rev1/4.eps} \\
+    \includegraphics*[width=70mm]{rev1/5.eps}
   %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps} 
-  \caption{Comparison of different samplings}%VH add  picture 
+  \caption{Comparison of different samplings with $\chi^2$ criterium (top) and $\Delta$ FWHM (bottom)}%VH add  picture 
    \label{samp}
 \end{figure}
-\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.
-For angle calculation and applying condition for first and final point we use  formula~\eqref{eq:lamb}.
-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.}
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=70mm]{new203/lin1.eps} \\
-    \includegraphics*[width=70mm]{new203/lin2.eps}
+{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.
+%For angle calculation and applying condition for first and final point was used  formula ~\ref{eq:lamb}.
+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.}
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{rev1/mmd5.eps} \\
+    \includegraphics*[width=70mm]{rev1/mmd10.eps}
   %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps} 
   \caption{Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.}%VH add  picture 
@@ -179 +194 @@
 
 \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). 
-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
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=90mm]{new203/histLINEAR.eps} 
-  
-  \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 
+%As the Lsmx configuration is physically impossible, 
+So 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
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=90mm]{rev1/hist1.eps} \\
+  \includegraphics*[width=90mm]{rev1/hist1fw.eps} 
+  \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 
    \label{biglin}
 \end{figure}
@@ -190 +206 @@
 
 %This section has been moved from elsewhere
-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.
-
-XXX Why you did not do the figure for FWHM ? XXX
-
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=70mm]{newfigures/Chi_Ndet.eps}
-  \caption{Effect of the sampling frequencies on the $\chi^2$. }
+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.
+
+
+
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{rev4/chN.eps}\\
+   \includegraphics*[width=70mm]{rev4/fwN.eps}
+  \caption{Effect of the sampling frequencies on the $\chi^2$ (top) and $\Delta$ FWHM (bottom). }
    \label{sampling_chi2}
 \end{figure}
-
+It can be seen at figure \ref{sampling_chi2} that beyond 33 sampling points the gain on the reconstructed $\chi^2$ is marginal.
 
 
@@ -206 +223 @@
 
 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}.
-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
+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. PCHIP interpolation has been chosen as it matches these requirements. 
 
 For low frequency extrapolation two methods have been investigated: Gaussian or Taylorian.
 
-In the Gaussian method, we define the extrapolation as follow:
+In the Gaussian method, was defined the extrapolation as follow:
 \begin{equation}
 \rho_{LF}(\omega)=Ae^{-(\omega-B)^2/2C^2}
@@ -233 +250 @@
 Approximation to the 4th order gives the following LF extrapolation:
 
-$$\rho_{LF}=|F(\omega)|=\sqrt{A+B\omega^2+C\omega^4}$$
+\begin{equation}
+\rho_{LF}=|F(\omega)|=\sqrt{A+B\omega^2+C\omega^4}
+\end{equation}
 
 Conditions for A, B and C constants are the same. Comparison of different LF extrapolation can be found on figure~\ref{lf}.
@@ -239 +258 @@
 \begin{figure}[!htb]
  \centering
-  \includegraphics*[width=65mm]{new203/LFsp.eps}\\
-  \includegraphics*[width=65mm]{new203/LFpr.eps}\\
-    \includegraphics*[width=65mm]{new203/LF.eps}
-    \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.     }
+  \includegraphics*[width=65mm]{rev2/lfsp.eps}\\
+  \includegraphics*[width=65mm]{rev2/lfpr.eps}\\
+
+    \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.}
    \label{lf}
 \end{figure}
 
-In the rest of this paper we use the Gaussian method.
+\begin{figure}[!htb]
+ \centering
+
+  \includegraphics*[width=65mm]{new203/LF.eps}
+    \includegraphics*[width=65mm]{rev3/lffw.eps}
+    \caption{Comparison of different LF extrapolation: histogram with mean $\chi^2$ for each method (top) and $\Delta$ FWHM (bottom).  }
+   \label{lf2}
+\end{figure}
+
+
+In the rest of this paper was used the Gaussian method.
 
 
 Several high frequency (HF) extrapolation method were also tested several of them. The most common~\cite{VBthesis,DESYthesis} is :
-$$\rho _ {HF} (\omega)=A\omega^{-4},$$ 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 
+\begin{equation}
+\rho _ {HF} (\omega)=A\omega^{-4},
+\end{equation}
+ 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 $\omega_f$.
 
 The second method use the same consideration as in Lai and Sievers~\cite{LaiS}: 
@@ -267 +299 @@
 \end{itemize}
 where $\omega_{fmax}$  is the last sampled point  of the spectrum. To satisfy the boundary condition two constants are needed, giving a two-terms extrapolation~:
-$$\rho_{HF}(\omega)=A\omega^{-2}+B\omega^{-3}$$
+\begin{equation}
+\rho_{HF}(\omega)=A\omega^{-2}+B\omega^{-3}
+\end{equation}
 or  extrapolation with the exponent as free parameter:
-$$\rho_{HF}(\omega)=A\omega^B$$
+\begin{equation}
+\rho_{HF}(\omega)=A\omega^B
+\end{equation}
 where the A and B coefficient are selected from the boundary conditions.
 Two other extrapolation methods have also been investigated:
@@ -283 +319 @@
 \begin{figure}[!htb]
  \centering
-  \includegraphics*[width=65mm]{new203/HFsp.eps}\\
-  \includegraphics*[width=65mm]{new203/HFprofile.eps}\\
-    \includegraphics*[width=65mm]{new203/HFGauss.eps}
-        \includegraphics*[width=65mm]{new203/HFLorenz.eps}
-    \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.}
+  \includegraphics*[width=65mm]{rev2/hfsp.eps}\\
+  \includegraphics*[width=65mm]{rev2/hfpr.eps}\\
+
+    \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.}
    \label{hf}
 \end{figure}
+\begin{figure}[!htb]
+ \centering
+    \includegraphics*[width=65mm]{new203/HFGauss.eps}\\
+        \includegraphics*[width=65mm]{rev4/hfgas.eps}\\
+
+    \caption{Comparison of different HF extrapolation for Gaussian~:histogram with mean $\chi^2$ (top) and $\Delta$ FWHM (bottom).}
+   \label{hf2}
+\end{figure}
+\begin{figure}[!htb]
+ \centering
+        \includegraphics*[width=65mm]{new203/HFLorenz.eps}\\
+          \includegraphics*[width=65mm]{rev4/hflor.eps}
+        \caption{Comparison of different HF extrapolation for Lorenzians~:histogram with mean $\chi^2$ (top) and $\Delta$ FWHM (bottom).}
+   \label{hf3}
+\end{figure}
+
+
 
 XXX If you show Lorenzian profiles for HF extrapolation why don't you also show it for LF extrapolation? XXX
@@ -297 +349 @@
 \section{Study of the reconstruction performance}
 
-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}.
+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}.
 
 \begin{figure}[!htb]
  \centering
 %  \includegraphics*[trim=0 0 275 0 ,clip,width=95mm]{plot1000700.png}
-  \includegraphics*[width=65mm]{new203/pic/541.eps}\\
-  \includegraphics*[width=65mm]{new203/pic/658.eps}\\
-   \includegraphics*[width=65mm]{new203/pic/914.eps}
+  \includegraphics*[width=65mm]{rev1/541.eps}\\
+  \includegraphics*[width=65mm]{rev1/658.eps}\\
+   \includegraphics*[width=65mm]{rev1/914.eps}
   \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.}
    \label{good_profiles}
@@ -312 +364 @@
  \centering
 %  \includegraphics*[trim=0 0 275 0 ,clip,width=95mm]{plot1000183.eps}
-  \includegraphics*[width=65mm]{new203/pic/227.eps}\\
-   \includegraphics*[width=65mm]{new203/pic/231.eps}\\
-    \includegraphics*[width=65mm]{new203/pic/667.eps}\\
+  \includegraphics*[width=65mm]{rev1/227.eps}\\
+   \includegraphics*[width=65mm]{rev1/231.eps}\\
+    \includegraphics*[width=65mm]{rev1/667.eps}\\
   \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.}
    \label{bad_profiles}
@@ -320 +372 @@
 
 
-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. 
-
-
-
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=70mm]{new203/pic/2.eps} \\
+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. 
+
+
+
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{rev1/2.eps} \\
   \includegraphics*[width=70mm]{new203/pic/3.eps} 
   \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
    \label{profiles_stats_hilbert}
 \end{figure}
-
-
+\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.}
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{rev3/compFig.eps}  
+  \caption{Explanation of $\chi^2$ distribution}% VH change name of picture and unite with other
+   \label{expKK}
+\end{figure}
 % VH add this block
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -338 +395 @@
  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.
  
-     XXX Figure \ref{mod}  needs to be fixed: 
-     Please use subplot to split the figure in 2 parts:
-     The upper 66\% of the figure should show the original and reconstructed profile.
-     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.
-     XXX 
+   
      
 \begin{figure}[!htb]
  \centering
-  \includegraphics*[width=70mm]{new203/pic/1.eps} 
+  \includegraphics*[width=70mm]{rev1/1.eps} 
   \caption{$\Delta_{FWXM}$  for 1000 profiles with both methods.}%VH add  picture 
    \label{fwxm}
@@ -352 +405 @@
 \begin{figure}[!htb]
  \centering
-  \includegraphics*[width=65mm]{new203/pic/6.eps}\\
-    \includegraphics*[width=65mm]{new203/pic/7.eps}
-    \caption{Original and reconstructed profile and their difference for two different profiles.}
+  \includegraphics*[width=65mm]{rev4/d1.eps}\\
+    \includegraphics*[width=65mm]{rev4/d2.eps}
+    \caption{Original and reconstructed profile and their difference for bad profile (top) and good profile (bottom).}
    \label{mod}
 \end{figure}
@@ -360 +413 @@
 
 
-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.
+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.
 
 \begin{figure}[!htb]
@@ -372 +425 @@
 
 % Effect of noise.
-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 :
-$$O_i' = O_i \times (1 + n_i) N_{max}$$
-
-XXX I think it should be XXX
-$$O_i' = O_i \times [1 + ( n_i N_{max}) ] $$
- 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 
+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 :
+\begin{equation}
+O_i' = O_i \times [1 + ( n_i N_{max}) ] 
+\end{equation}
+ 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 
 maximum 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.
  
@@ -393 +445 @@
 XXX To be updated XXX
 
-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.
+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.