Changeset 165 in ETALON for reconstruction

Timestamp:

Mar 2, 2015, 2:09:39 PM (9 years ago)

Author:

hodnevuc

Message:

new version

Location:

reconstruction/long_paper3

Files:

• phase_reconstruction_paper.aux (modified) (2 diffs)
• phase_reconstruction_paper.log (modified) (7 diffs)
• phase_reconstruction_paper.pdf (modified) (previous)
• phase_reconstruction_paper.synctex.gz (modified) (previous)
• phase_reconstruction_paper.tex (modified) (7 diffs)

reconstruction/long_paper3/phase_reconstruction_paper.aux

@@ -10 +10 @@
 \@writefile{toc}{\contentsline {section}{Longitudinal bunch profile measurement at particle accelerators}{1}}
 \@writefile{toc}{\contentsline {section}{Reconstruction methods}{1}}
-\@writefile{toc}{\contentsline {section}{description of the simulations}{1}}
+\@writefile{toc}{\contentsline {section}{Description of the simulations}{1}}
 \citation{E203prstab}
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Example $\chi ^2$ defect.\relax }}{2}}
+\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
+\newlabel{Offsine}{{1}{2}}
+\newlabel{eq:lamb}{{2}{2}}
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Comparison of different samplings\relax }}{2}}
+\newlabel{samp}{{2}{2}}
 \citation{pchip}
+\citation{VBthesis}
+\citation{VBthesis}
+\citation{DESYthesis}
 \citation{LaiS}
-\citation{VBthesis}
-\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Comparison of different interpolations\relax }}{2}}
-\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
-\newlabel{inter}{{1}{2}}
-\@writefile{toc}{\contentsline {section}{Study of the reconstruction performance}{2}}
-\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Comparison of different exptrapolations\relax }}{3}}
-\newlabel{extra}{{2}{3}}
-\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Example 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 }}{3}}
-\newlabel{good_profiles}{{3}{3}}
-\@writefile{toc}{\contentsline {section}{Stability of reconstruction}{3}}
-\@writefile{lof}{\contentsline {figure}{\numberline {4}{\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 }}{4}}
-\newlabel{bad_profiles}{{4}{4}}
-\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Explanation\relax }}{4}}
-\newlabel{Offsine}{{5}{4}}
-\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces {$\Delta _{FWHM}$ (top) and $\chi ^2$ (bottom) distribution of our 1000 simulations reconstructed using the Hilbert transform method and Kramers-Kronig reconstruction method.}\relax }}{4}}
-\newlabel{profiles_stats_hilbert}{{6}{4}}
-\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Explanation ...\relax }}{4}}
-\newlabel{mod}{{7}{4}}
+\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.\relax }}{3}}
+\newlabel{lin12}{{3}{3}}
+\@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}}
+\newlabel{biglin}{{4}{3}}
+\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Comparison of different LF interpolation:example of spectrum and profile, histo with mean $\chi ^2$ for comparison. Gauss and Teylor are method explained above and Real -is replacement LF part of extrapolation by real spectrum. \relax }}{4}}
+\newlabel{lf}{{5}{4}}
+\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Comparison of different HF interpolation:example of spectrum and profile, histo with mean $\chi ^2$ for comparison for Gaussians and Lorenzians.\relax }}{4}}
+\newlabel{hf}{{6}{4}}
+\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Example 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}}
+\newlabel{good_profiles}{{7}{5}}
+\@writefile{toc}{\contentsline {section}{Study of the reconstruction performance}{5}}
+\@writefile{lof}{\contentsline {figure}{\numberline {8}{\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 }}{5}}
+\newlabel{bad_profiles}{{8}{5}}
 \citation{Pelliccia:2014vba}
-\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces $\Delta _{FWXM}$ for 1000 profiles with both methods.\relax }}{5}}
-\newlabel{fwxm}{{8}{5}}
-\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Comparison of different samplings\relax }}{5}}
-\newlabel{sampl}{{9}{5}}
-\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Effect of the sampling frequencies on the $\chi ^2$. \relax }}{5}}
-\newlabel{sampling_chi2}{{10}{5}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Effect of scaling the constraints on the parameters $\sigma _i$ and $\mu _i$ on the $\chi ^2$.\relax }}{5}}
-\newlabel{sigma_chi2}{{11}{5}}
+\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces {$\Delta _{FWHM}$ (top) and $\chi ^2$ (bottom) distribution of our 1000 simulations reconstructed using the Hilbert transform method and Kramers-Kronig reconstruction method.}\relax }}{6}}
+\newlabel{profiles_stats_hilbert}{{9}{6}}
+\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces $\Delta _{FWXM}$ for 1000 profiles with both methods.\relax }}{6}}
+\newlabel{fwxm}{{10}{6}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Explanation to figure (\ref  {fwxm})\relax }}{6}}
+\newlabel{mod}{{11}{6}}
+\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Effect of the sampling frequencies on the $\chi ^2$. \relax }}{6}}
+\newlabel{sampling_chi2}{{12}{6}}
 \bibcite{OTR_LURE}{1}
 \bibcite{ODR_Cianchi}{2}
@@ -50 +54 @@
 \bibcite{pchip}{8}
 \bibcite{LaiS}{9}
-\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Distribution of the $\chi ^2$ in the case of a lorenzian distribution.\relax }}{6}}
-\newlabel{lorenz}{{12}{6}}
-\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Mean $\chi ^2$ as function of noise amplitude.\relax }}{6}}
-\newlabel{noise}{{13}{6}}
-\@writefile{toc}{\contentsline {section}{Discussion}{6}}
+\bibcite{DESYthesis}{10}
+\@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}}
+\newlabel{sigma_chi2}{{13}{7}}
+\@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces Distribution of the $\chi ^2$ in the case of a lorenzian distribution.\relax }}{7}}
+\newlabel{lorenz}{{14}{7}}
+\@writefile{toc}{\contentsline {section}{Discussion}{7}}
+\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Mean $\chi ^2$ as function of noise amplitude.\relax }}{7}}
+\newlabel{noise}{{15}{7}}

reconstruction/long_paper3/phase_reconstruction_paper.log

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reconstruction/long_paper3/phase_reconstruction_paper.tex

@@ -81 +81 @@
 
 
-To recover the phase from the amplitude, the function should  be written as: $log(\varepsilon(\omega))=log(\rho(\omega))+i\Theta(\omega)$ with $\rho(\omega)$ its amplitude and $\Theta(\omega)$ its phase. 
+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:
 $$\Theta(\omega_0)  =  \frac{2\omega_0}{\pi} \textit{P}\int^{+ \infty}_{0}\frac{ln(\rho(\omega) )}{\omega_0^2-\omega^2}d\omega$$
@@ -89 +89 @@
 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 ($\textit{F}$): $$\textit{F}(\textit{H}(u))(\omega)=(-isgn(\omega))\textit{F}(u)(\omega),$$ the calculation of phase can use optimised FFT code and is much faster than  calculating the Kramers-Kronig's integral.}%VH add text
+{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.
 
-\section{description of the simulations}
+\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.
@@ -102 +102 @@
 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).
 
-Different distributions have been used for the frequencies $\omega_i$: linear, logarithmic, triple-sine,E-203+linear.%,derivative method, method of useful information. 
-In most sampling schemes 33 frequencies were used to make it comparable with Triple-sine. More detail about each:
-\begin{itemize}
-
-\item \textit{Linear sampling} There sampling points distributed uniformly. Fist point of sampling is first point in spectrum, last point -- is last point in Triple-sine sampling. To get points from spectrum, we use formula:
-$$round(Interval/Np*(1:Np))-\alpha,$$ where $Interval$ -- is samling interval in spectrum, $Np$ -- is number of points, $\alpha$ -- is initial displacement that helps to avoid low frequency extrapolation
-\item \textit{Logarithmic sampling}. Point is distributed according logarithmic low. For this, we use $$round(e^{\beta \times (1:Np)}),$$ where $\beta$ -- is constant, that set interval of sampling.
-\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:
-$$\frac{c}{\omega_i}=l_n (1-cos(\Theta_i ))$$ with $l_n =50, 250, 1500 \mu m$ and $\Theta$ varying between $40^o$ and $140^o$.
-\item \textit{Triple-sine+Linear}.
- There is linear sampling on three different pieces (Triple-sine), that determined by $l_n$ and on each piece is it own linear sampling.
-%\item \textit{Derivative method}. This method have some preliminary studies. Considerations was next: detectors should be put where the greatest variety of changes. Restore gentle  areas is easier than downhill. So was generated  1000 profiles. From each profile was taken derivative and then  sum of modules of it determine desired distribution.
-%\item \textit{Method of useful information}. This method also have some preliminary studies. Considerations was next: It's easy to recover gaussian (which also have gaussian spectrum), but if we want to see a more complex shape that is different from gaussian, we must to see a spectrum more details. So 1000 spectrum profiles was fitted by gaussian and then  sum of modules of it determine desired distribution.
-\end{itemize}
-%This would have to be detailed in a longer paper.
-
-{After applying the sampling procedure we need to interpolate and extrapolate the data to have a larger number of samples. Interpolation is done using Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)\cite{pchip}. This finds values of an underlying interpolating function P(x) at intermediate points, such that:}
-\begin{itemize}
-\item On each subinterval $x_k\leq x\leq x_{k+1},$   is the cubic Hermite interpolant to the given values and certain slopes at the two endpoints.
-\item First derivative $\frac{dP}{dx}$ is continuous.
-\item The slopes at the $x_j$ are chosen in such a way that   preserves the shape of the data and respects monotonicity. 
-\end{itemize}
-For extrapolation I use
-$$\rho _ {HF} (\omega)=A\omega^{-4},$$ where $\rho$ -- is spectrum and $A=\rho_H \omega_H^{4} $.\\
-Some confirmation of extrapolation by this formula can be found in \cite{LaiS}.
-To find the high asymptote we make use of finite size of the electron bunch. Let its two end points be at $z=0,\sigma_z$ then $S(0)=S(\sigma_z)=0$. Integrating the spectral function by parts over the nonzero range gives. 
-$$\widehat{S}(\omega)=\int_0^\infty dzS(z)e^{i(\frac{\omega}{c})z}=\frac{S(z)}{i\frac{\omega}{c}} 
-e^{i(\frac{\omega}{c})z} \Big|_0^{\sigma_z}-\frac{S^{\prime}(z)}{\Big(i\frac{\omega}{c}\Big)^2} 
-e^{i(\frac{\omega}{c})z} \Big|_0^{\sigma_z}+\ldots$$
-Since the first term vanishes because of the boundary conditions, the leading term is proportional to $\omega^{-2}$; hence for large $\omega$, $F(\omega)$ varies as $\omega^{-4}$.\\
-
-Before, we tested all avaible in matlab interpolation (pchip, spline, linear, nearest) but but choose pchip as most appropriate (\ref{inter}). 
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=65mm]{newFig/interSpec.eps}\\
-  \includegraphics*[width=65mm]{newFig/interProf.eps}
-    \caption{Comparison of different interpolations}
-   \label{inter}
-\end{figure}
-Also was tested exponetial HF exrtapolation (($Ae^{-\alpha \omega^2}$,$Ae^{-\alpha \omega}$)), but they works well only for gaussian-like profiles, for more complicated profiles four degree polinimial are better\cite{VBthesis} as shown at figure (\ref{extra}).
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=65mm]{newFig/extraSpectr.eps}\\
-    \includegraphics*[width=65mm]{newFig/exptraProf.eps}
-    \caption{Comparison of different exptrapolations}
-   \label{extra}
-\end{figure}
-
- {And than we applied our reconstruction techniques to reconstruct the original profile.}% VH add and change text
-As can be expected in some cases the reconstruction went very well and in some other cases it was not as convincing. An example of a well reconstructed profile is shown on figure~\ref{good_profiles} and examples of poorly reconstructed profile is shown on figure~\ref{bad_profiles}.
-
-\begin{figure}[!htb]
- \centering
-%  \includegraphics*[trim=0 0 275 0 ,clip,width=95mm]{plot1000700.png}
-  \includegraphics*[width=65mm]{newfigures/GRAPH/good.eps}\\
-  \includegraphics*[width=65mm]{newfigures/new2/good1.eps}\\
-   \includegraphics*[width=65mm]{newfigures/new2/good2.eps}
-  \caption{Example 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}
-\end{figure}
-
-\begin{figure}[!htb]
- \centering
-%  \includegraphics*[trim=0 0 275 0 ,clip,width=95mm]{plot1000183.eps}
-  \includegraphics*[width=65mm]{newfigures/GRAPH/bad.eps}\\
-   \includegraphics*[width=65mm]{newfigures/new2/bad1.eps}\\
-    \includegraphics*[width=65mm]{newfigures/new2/bad2.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}
-\end{figure}
-
-
-
-\section{Study of the reconstruction performance}
-{To estimate the performance of the reconstruction several estimators are available. We choose to use the $\chi^2$, defined as follow:
+To estimate the performance of the reconstruction several estimators are available. We choose to use the $\chi^2$, defined as follow:
 $$\chi^2=\sum_i\omega_i^2(O_i-E_i)^2/N,$$
-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. As we deal with the module of the Fourier transform, we lose the position and direction information of the profile, therefore we first minimize this estimator to find the correct position and direction.} % VH change and add text
-However for two very similar profiles but with a slight offset, a bad $\chi^2$ will be returned (look at figure \ref{Offsine}).
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=70mm]{newFig/offset_sine.eps}
-  \caption{Explanation}% VH change name of picture and unite with other
+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 for two very similar profiles but with a slight offset, will give worse $\chi^2$ than oscilated one (look at figure \ref{Offsine}).
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{a.eps}
+  \caption{Example $\chi^2$ defect.}% VH change name of picture and unite with other
    \label{Offsine}
 \end{figure}
@@ -196 +122 @@
 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.
 
+
+
+Different distributions have been used for the frequencies $\omega_i$: linear, logarithmic, triple-sine. 
+In most sampling schemes 33 frequencies were used to make it comparable with Triple-sine. More detail about each:
+\begin{itemize}
+\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:
+\begin{equation} \label{eq:lamb}
+\frac{c}{\omega_i}=l_n (1-cos(\Theta_i )) 
+\end{equation}
+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:
+$$\omega_0*exp(log(\omega_f/\omega_0)\times(0:32)/32).$$
+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{Sampling study is important, as it will tell to us the best position for detectors and will help to optimize system. Linear sample spectrum gives the best result (see figure (\ref{samp})). 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]{newFig/lin27e203line203.eps} 
+  \caption{Comparison of different samplings}%VH add  picture 
+   \label{samp}
+\end{figure}
+\textbf{However, this is the ideal case. In fact detector systems have spacial dimensions (10 degrees) and there is also a limit on the start and end points of detectors location (35-145 degrees). So linear sampling at a wide range of frequencies is imposible with this number of points. So we make study to investigate how many linear sampled point we can achived with given angle difference between detectors.
+For angle calculation and applying condition for first and final point we use  formula ~\eqref{eq:lamb}.
+On figure (\ref{lin12}) examples of detector position is presented. Red points is points calculated by formula and blue is possible position of detector, which does not break minimum detector distance (MDD).}
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{new203/lin1.eps} \\
+    \includegraphics*[width=70mm]{new203/lin2.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 
+   \label{lin12}
+\end{figure}
+
+\textbf{Our aim was to see if it possible to reduce number of detectors but save perfomance (at least not worse Triple-sine). But as show at figure (\ref{biglin}) it is impossible due to geometrical consideration.}\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 
+   \label{biglin}
+\end{figure}
+\textbf{So at that moment Triple-sine is best sapmling method and we will use it further in our simulation.}
+
+
+
+
+
+{After applying the sampling procedure we need to interpolate and extrapolate the data to have a larger number of points in spectrum. Interpolation is done using Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)\cite{pchip}, as in \cite{{VBthesis}} \textbf{ For interpolation it use next criteria, to how function must look like in result. On each interpolation subinterval it use cubic Hermite interpolant to sapmling values and for maximum save slopes at the two endpoints, so first derivative of spectrum is continuous, it preserves the shape of the data and respects monotonicity.}
+
+For (LF) interpolation we have tested two methods. First of them is gaussian:
+\begin{equation}
+\rho_{LF}(\omega)=Ae^{-(\omega-B)^2/2C^2}
+\end{equation}
+Where constants A,B,C was choosen from next condition:
+\begin{itemize}
+\item $\rho_{LF}(0)=1$
+\item $\rho_{LF}(\omega_0)=\rho(\omega_0)$
+\item $\rho_{LF}'(\omega_0)=\rho'(\omega_0)$
+\end{itemize}
+\textbf{We get it from consideration, that according central limit theorem in time space we will get something Gaussian-like and from this in frequency space also will be Gaussian.}\par \textbf{A different approach based at scheduling in a row an exponent in integral:}
+\begin{multline}
+F(\omega)=\int_0^\infty dtS(t)e^{-i(\omega t)}
+=\int_0^\infty dtS(t) \sum_{k=0}^{\infty}\frac{(-i\omega t)^k}{k!}=\\
+\sum_{k=0}^{\infty} \left(\frac{(-i\omega)^k}{k!} \int_0^\infty dtS(t)t^k\right)
+=\sum_{k=0}^{\infty} \left(\frac{(-i\omega)^k}{k!} <t^k>\right)
+\end{multline} 
+\textbf{Schedule to 4th order and taking module of this will give us approach to LF extrapolation:}
+
+$$\rho_{LF}=|F(\omega)|=\sqrt{A+B\omega^2+C\omega^4}$$
+
+\textbf{Conditions for A,B,C coefficients are the same. Comparison of different LF exptrapolation can be finded at figure (\ref{lf}).For this simulation we use Hilbert method of phase recovery and $A\omega^B$ high frequency extrapolation. For next simulation we use Gauss method.}
+\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 interpolation:example of spectrum and profile, histo with mean $\chi^2$ for comparison. Gauss and Teylor are method explained above and Real -is replacement LF part of extrapolation by real spectrum. }
+   \label{lf}
+\end{figure}
+
+\textbf{For high frequency (HF) extrapolation we tested several of them. First is most common (\cite{VBthesis},\cite{DESYthesis}):}
+$$\rho _ {HF} (\omega)=A\omega^{-4},$$ where $\rho$ -- is spectrum and $A=\rho_H \omega_H^{4} $.\\
+\textbf{For second we will use same consideration as Lai and Sievers(\cite{LaiS}).
+Assume that bunch size is finite and two end points be at $z=0,\sigma_z$ then longditudinal charge distribution $S(0)=S(\sigma_z)=0$. Let`s take form factor integral by parts:}
+\begin{multline}
+F(\omega)=\int_0^\infty dzS(z)e^{i(\frac{\omega}{c})z}=\\
+=\frac{S(z)}{i\frac{\omega}{c}} 
+e^{i(\frac{\omega}{c})z} \Big|_0^{\sigma_z}-\frac{S^{\prime}(z)}{\Big(i\frac{\omega}{c}\Big)^2} 
+e^{i(\frac{\omega}{c})z} \Big|_0^{\sigma_z}+\ldots
+\end{multline} 
+\textbf{First term vanishes because of the boundary conditions, so for big $\omega$, $F(\omega)$ proportional to $\omega^{-2}$. For HF extrapolation we have two condition:}
+\begin{itemize}
+\item $\rho_{HF}(\omega_f)=\rho(\omega_f)$
+\item $\rho_{HF}'(\omega_f)=\rho'(\omega_f)$
+\end{itemize}
+\textbf{where $\omega_f$ -- is last point in spertrum. So for satisfaction of boundary condition we need at least two constant. So we also tested extrapolation with two terms:}
+$$\rho_{HF}(\omega)=A\omega^{-2}+B\omega^{-3}$$
+\textbf{and one extrapolation without fixed power:}
+$$\rho_{HF}(\omega)=A\omega^B$$
+\textbf{where A and B coefficient are selected from boundary conditions. Except this three we tested zero exptratolation (just replacing HF part by zero) and compare all of them with case where HF part replaced by real spectrum. Results can be finded at figure (\ref{hf}).For this simulation we use Hilbert method of phase recovery and Gauss LF extrapolations.}
+\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 interpolation:example of spectrum and profile, histo with mean $\chi^2$ for comparison for Gaussians and Lorenzians.}
+   \label{hf}
+\end{figure}
+
+
+\textbf{ At this procedure of spectrum recovery is finished and than we applied our reconstruction techniques to reconstruct the original profile.}
+As can be expected in some cases the reconstruction went very well and in some other cases it was not as convincing. An example of a well reconstructed profile is shown on figure~\ref{good_profiles} and examples of poorly reconstructed profile is 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}
+  \caption{Example 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}
+\end{figure}
+
+\begin{figure}[!htb]
+ \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}\\
+  \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}
+\end{figure}
+
+
+
+\section{Study of the reconstruction performance}
+{ As we deal with the module of the Fourier transform, we lose the position and direction information of the profile, therefore we first minimize $\chi^2$ to find the correct position and direction.} % VH change and add text
+
 {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 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.}% VH changed text
 
 \begin{figure}[!htb]
  \centering
-  \includegraphics*[width=70mm]{newfigures/FWHM.eps} \\
-  \includegraphics*[width=70mm]{newfigures/chi2.eps} 
+  \includegraphics*[width=70mm]{new203/pic/2.eps} \\
+  \includegraphics*[width=70mm]{new203/pic/3.eps} 
   \caption{{$\Delta_{FWHM}$  (top)  and $\chi^2$ (bottom) distribution of our 1000 simulations reconstructed using the Hilbert transform method and Kramers-Kronig reconstruction method.}}% VH change name of picture and unite with other
    \label{profiles_stats_hilbert}
@@ -211 +284 @@
 \begin{figure}[!htb]
  \centering
-  \includegraphics*[width=65mm]{newFig/goodRec.eps}\\
-    \includegraphics*[width=65mm]{newFig/badRec.eps}
-    \caption{Explanation ...}
-   \label{mod}
-\end{figure}
-
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=70mm]{newfigures/FWXM.eps} 
+  \includegraphics*[width=70mm]{new203/pic/1.eps} 
   \caption{$\Delta_{FWXM}$  for 1000 profiles with both methods.}%VH add  picture 
    \label{fwxm}
 \end{figure}
-{Linear selection works best as interpolation procedure that is present in the recovery range, gives the best result in that case. As You see from figure \ref{sampl} even with 27 point linear sapmling works quite good.}% VH add text
-\begin{figure}[!htb]
- \centering
-  \includegraphics*[width=70mm]{newFig/linloge203.eps} \\
-  \includegraphics*[width=70mm]{newFig/lin27e203line203.eps} 
-  \caption{Comparison of different samplings}%VH add  picture 
-   \label{sampl}
-\end{figure}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%\begin{figure}[!htb]
-% \centering
-%  \includegraphics*[width=70mm]{THPME088f5.eps} \\
-%  \includegraphics*[width=70mm]{THPME088f6.eps} 
-%  \caption{$\Delta_{FWXM}$   (top)  and $\chi^2$   (bottom)  distribution of our 1000 simulations reconstructed using the Kramers-Kronig reconstruction method.}
-%   \label{profiles_stats_KK}
-%\end{figure}
-
-\section{Stability of reconstruction}
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=65mm]{new203/pic/6.eps}\\
+    \includegraphics*[width=65mm]{new203/pic/7.eps}
+    \caption{Explanation to figure (\ref{fwxm})}
+   \label{mod}
+\end{figure}
+
+
 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 different sampling ranging from 5 to 120 spectrum points. The effect of changing the sampling frequencies on the  $\chi^2$ is shown on figure~\ref{sampling_chi2}.\textbf{Fow this study we use linear sampling with differnt number of points in it, 1000 profiles for each point and Hilbert reconstruction algorithm.}
 
@@ -338 +392 @@
 \newblock {Determination of the Time Profile of Picosecond-Long Electron Bunches through the use of Coherent Smith-Purcell Radiation}.
 \newblock 2008.
+
 \bibitem{pchip}
 MATLAB documentation
@@ -348 +403 @@
 \newblock { Phys. Rev. E, Vol. 50, No. 6, R4294 } 1994
 
+\bibitem{DESYthesis}
+Lars Frohlich
+\newblock {Bunch Length Measurements Using a Martin-Puplett Interferometer at the VUV-FEL}.
+\newblock 2005.
+