Changeset 199 in ETALON for reconstruction

Timestamp:

Mar 11, 2015, 6:08:55 PM (9 years ago)

Author:

delerue

Message:

MOore updates

Location:

reconstruction/long_paper3

Files:

• phase_reconstruction_paper.aux (modified) (4 diffs)
• phase_reconstruction_paper.log (modified) (18 diffs)
• phase_reconstruction_paper.pdf (modified) (previous)
• phase_reconstruction_paper.synctex.gz (modified) (previous)
• phase_reconstruction_paper.tex (modified) (6 diffs)

reconstruction/long_paper3/phase_reconstruction_paper.aux

@@ -24 +24 @@
 \citation{pchip}
 \citation{VBthesis}
-\citation{VBthesis}
-\citation{DESYthesis}
-\citation{LaiS}
+\citation{VBthesis,DESYthesis}
 \@LN@col{1}
 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.\relax }}{3}}
@@ -33 +31 @@
 \@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}}
+\citation{LaiS}
 \@LN@col{1}
-\@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}}
+\@writefile{lof}{\contentsline {figure}{\numberline {5}{\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}}
 \newlabel{lf}{{5}{4}}
 \@LN@col{2}
-\@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{toc}{\contentsline {section}{Study of the reconstruction performance}{4}}
+\citation{Pelliccia:2014vba}
 \@LN@col{1}
+\@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 }}{5}}
+\newlabel{hf}{{6}{5}}
+\@LN@col{2}
 \@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}}
-\@LN@col{2}
-\@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}
-\@LN@col{1}
-\@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}}
-\@LN@col{2}
-\@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}}
+\@writefile{toc}{\contentsline {section}{Discussion}{5}}
 \bibcite{OTR_LURE}{1}
 \bibcite{ODR_Cianchi}{2}
@@ -63 +51 @@
 \bibcite{E203prstab}{5}
 \bibcite{Pelliccia:2014vba}{6}
+\@LN@col{1}
+\@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 }}{6}}
+\newlabel{bad_profiles}{{8}{6}}
+\@LN@col{2}
+\@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}}
 \bibcite{VBthesis}{7}
 \bibcite{pchip}{8}
@@ -68 +64 @@
 \bibcite{DESYthesis}{10}
 \@LN@col{1}
+\@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{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Mean $\chi ^2$ as function of noise amplitude.\relax }}{8}}
+\newlabel{noise}{{15}{8}}
 \@LN@col{2}

reconstruction/long_paper3/phase_reconstruction_paper.log

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reconstruction/long_paper3/phase_reconstruction_paper.tex

@@ -165 +165 @@
 \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
+\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
@@ -173 +174 @@
    \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:
+
+
+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
+
+For low frequency extrapolation two methods have been investigated: Gaussian or Taylorian.
+
+In the Gaussian method, we define the extrapolation as follow:
 \begin{equation}
 \rho_{LF}(\omega)=Ae^{-(\omega-B)^2/2C^2}
 \end{equation}
-Where constants A,B,C was choosen from next condition:
+Where  $\rho_ {HF} (\omega)$ is the extrapolated spectrum at low frequency and the constants A, B, and C were chosen from the following conditions:
 \begin{itemize}
 \item $\rho_{LF}(0)=1$
@@ -191 +191 @@
 \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:}
+
+The extrapolation relies in the fact that according to the central limit theorem in the time space the expected profile is Gaussian-like and in the frequency space it will also be Gaussian.
+
+The other extrapolation method is based on Taylor expansion with the following definition:
 \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!} \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:}
+
+Approximation to the 4th order gives the following 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.}
+Conditions for A, B and C constants are the same. Comparison of different LF extrapolation can be found in figure~\ref{lf}.
+
 \begin{figure}[!htb]
  \centering
@@ -208 +213 @@
   \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. }
+    \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.     }
    \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:}
+In the rest of this paper we use 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 
+
+The second method use the same consideration as in Lai and Sievers~\cite{LaiS}: 
+Assuming that the bunch size is finite with two end points at $z=0$ and at $z=\sigma_z$ then the longitudinal charge distribution ($S$) must follow $S(0)=S(\sigma_z)=0$. 
+An integration by parts gives~:
 \begin{multline}
 F(\omega)=\int_0^\infty dzS(z)e^{i(\frac{\omega}{c})z}=\\
@@ -222 +232 @@
 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:}
+The first term vanishes because of the boundary conditions, so for large $\omega$, $F(\omega)$ is proportional to $\omega^{-2}$ and two conditions have to be matched~:
 \begin{itemize}
-\item $\rho_{HF}(\omega_f)=\rho(\omega_f)$
-\item $\rho_{HF}'(\omega_f)=\rho'(\omega_f)$
+\item $\rho_{HF}(\omega_{fmax})=\rho(\omega_{fmax})$
+\item $\rho_{HF}'(\omega_{fmax})=\rho'(\omega_{fmax})$
 \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:}
+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~:
 $$\rho_{HF}(\omega)=A\omega^{-2}+B\omega^{-3}$$
-\textbf{and one extrapolation without fixed power:}
+or  extrapolation with the exponent as free parameter:
 $$\rho_{HF}(\omega)=A\omega^B$$
-\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.}
+where the A and B coefficient are selected from the boundary conditions.
+Two other extrapolation methods have also been investigated:
+\begin{itemize}
+\item $\rho_{HF}(\omega_f)= 0 $ for $ \omega_f >  \omega_{fmax}$
+\item $\rho_{HF}(\omega_f)= $\rho_{real}(\omega_f) $ for $ \omega_f >  \omega_{fmax} $ where $\rho_{real}$ is the real spectrum.
+\end{itemize}
+
+These extrapolation methods are compared in figure~\ref{hf}.
+
+ 
+ 
 \begin{figure}[!htb]
  \centering
@@ -241 +261 @@
    \label{hf}
 \end{figure}
+xxx For this simulation we use Hilbert method of phase recovery and Gauss LF extrapolations.}
+
+