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7466ae1c | 1 | \documentclass[11pt]{article} |
2 | \usepackage[margin=2cm,twoside]{geometry} | |
3 | \usepackage{amsmath} | |
4 | \usepackage{amstext} | |
5 | \usepackage{units} | |
6 | \usepackage{graphicx} | |
7 | \title{Calculation of signals in the FMD simulation and reconstruction} | |
8 | \author{Christian Holm Christensen} | |
9 | \date{\today} | |
10 | \def\MeV#1{\unit[#1]{MeV}} | |
11 | \def\N#1{\unit[#1]{N}} | |
12 | \def\Q#1{\unit[#1]{Q}} | |
13 | \def\ALTRO{{\scshape altro}} | |
14 | \def\RCU{{\scshape rcu}} | |
15 | \def\FMD{{\scshape fmd}} | |
16 | \def\VA{{\scshape va1}} | |
17 | \def\class#1{{\small\ttfamily #1}} | |
18 | \def\DeltaMip{\ensuremath{\bar{\Delta}_{mip}}} | |
19 | \begin{document} | |
20 | \maketitle | |
21 | ||
22 | \section*{Introduction} | |
23 | ||
24 | This is meant as a reminder of what kind of manipulations we do in the | |
25 | simulation and reconstruction of \FMD{} data. Please refer to | |
26 | \tablename~\ref{tab:conventions} for conventions and constants used in | |
27 | this document. | |
28 | ||
29 | \begin{table}[Htbp] | |
30 | \begingroup | |
31 | \centering | |
32 | \begin{tabular}{|l|c|r|p{.6\textwidth}|} | |
33 | \hline | |
34 | \textbf{Symbol} & \textbf{Unit} & \textbf{Value} & \textbf{Description}\\ | |
35 | \hline | |
36 | $\delta_{ij}$ & \MeV{} & & Energy loss by particle $j$ in strip $i$\\ | |
37 | $\Delta_i$ & \MeV{} & & Summed energy loss in strip $i$\\ | |
38 | ${}_{mc}$ & & & Monte-Carlo mark\\ | |
39 | $q_{mip}$ & \Q{} & 29.67 & Number of $e^-$ charges for a MIP\\ | |
40 | $\DeltaMip{}$ & \MeV{} & 0.124 & Average energy deposition by a MIP\\ | |
41 | $c_i$ & \N{} & $[0,10^2-1]$ & ADC counts in strip $i$\\ | |
42 | $g_i$ & \N{}/\Q{} & $~2.2$ & Pulser calibrated gain for strip $i$\\ | |
43 | $p_i$ & \N{} & $~100$ & Pedestal value in strip $i$\\ | |
44 | $n_i$ & \N{} & $2-4$ & Noise value of strip $i$\\ | |
45 | $f_{ol}$ & & 4 & On--line noise suppression factor\\ | |
46 | $f_{reco}$ & & 4 & Reconstruction noise suppression | |
47 | factor\\ | |
48 | $b$ & & 6 & Shaping time parameter\\ | |
49 | $\rho$ & \unit[g\,cm\textsuperscript{-3}] & 2.33 & Density of silicon\\ | |
50 | $T$ & \unit[cm] & 0.032 & Thickness of sensors\\ | |
51 | \hline | |
52 | \end{tabular} | |
53 | \endgroup | |
54 | \vspace{1ex} | |
55 | \par | |
56 | \noindent Here, | |
57 | \begin{align*} | |
58 | \DeltaMip{} &= \unit[1.664]{MeV cm^2 g^{-1}} \rho\,T\\ | |
59 | &= \unit[1.664]{MeV cm^2 g^{-1}} \unit[2.33]{g\,cm^{-3}} | |
60 | \unit[0.032]{cm} \\ | |
61 | &= \MeV{0.124} | |
62 | \end{align*} | |
63 | where $\rho=\unit[2.33]{g\,cm^{-3}}$ is the density of silicon, and | |
64 | $T=\unit[320]{\mu{}m}$ the thickness of the silicon sensor. | |
65 | ||
66 | The factor $q_{mip}$ is given by the electronics of the front--end | |
67 | cards of the \FMD{} and was measured in the laboratory in August 2008. | |
68 | It is a digital--to--analogue setting corresponding to 1 MIP in the | |
69 | pulser injection circuit on the front--end electronics. | |
70 | \caption{Conventions used in this document, and constant values.} | |
71 | \label{tab:conventions} | |
72 | \end{table} | |
73 | ||
74 | ||
75 | \section*{Simulations} | |
76 | ||
77 | In the hits (\class{AliFMDHit}) are generated per strip for each | |
78 | particle that impinges on a strip. Stored in the hit are the energy | |
79 | loss $\delta_{i,mc}$ of the particle impinging as well as the path | |
80 | length $l_{i,mc}$ of the particle track through the strip. | |
81 | ||
82 | When generating simulated detector signal (\class{AliFMDSDigit} or | |
83 | \class{AliFMDDigit}) the energy loss in all hits in a single strip is | |
84 | summed to a total energy loss in the strip. | |
85 | \begin{equation} | |
86 | \label{eq:sim:sum_eloss} | |
87 | \Delta_{i,mc} = \sum_j \delta_{ij,mc} \quad[\MeV{}] | |
88 | \end{equation} | |
89 | ||
90 | The detector signal (ADC counts) is then calculated using the fixed | |
91 | gain of the \VA{} pre-amplifiers ($q_{mip}$), the average | |
92 | energy deposition of a MIP $\DeltaMip{}$, and the pulser calibrated | |
93 | gain of the strip $g_i$. These numbers combine to a conversion | |
94 | factor $f_{i,mc}$ given by | |
95 | ||
96 | \begin{equation} | |
97 | \label{eq:sim:conversion_factor} | |
98 | f_{i,mc} = \frac{q_{mip} g_i}{\DeltaMip{}} \quad [\N{} \MeV{}^{-1}] | |
99 | \end{equation} | |
100 | ||
101 | This factor and the constant value $C_i$ is then used to calculate the | |
102 | number of ADC counts | |
103 | ||
104 | \begin{equation} | |
105 | \label{eq:sim:adc_counts} | |
106 | c_i = \Delta_{i,mc} f_{i,mc} + C_i \quad[\N{}] | |
107 | \end{equation} | |
108 | ||
109 | In case of multiple samples ($r$) of the same strip, each sample $j$ is | |
110 | given by | |
111 | \begin{equation} | |
112 | \label{eq:sim:sub_adc_counts} | |
113 | c_{ij,mc} = f_{i,mc} \left(\Delta_{i,mc} + (\Delta_{i-1,mc}-\Delta_i) e^{-b | |
114 | \frac{j}{r}}\right)+C_i \quad[\N{}] | |
115 | \end{equation} | |
116 | where $j$ runs from 1 to $r$ (the number of samples), and $b$ is a | |
117 | constant that depends on the shaping time of the \VA{} | |
118 | pre-amplifier (see also \figurename~\ref{fig:sim:va1_response}). | |
119 | ||
120 | \begin{figure}[Htbp] | |
121 | \centering | |
122 | \includegraphics[keepaspectratio,width=.45\textwidth]{va1_response} | |
123 | \includegraphics[keepaspectratio,width=.45\textwidth]{va1_train} | |
124 | \caption{Left: Shaping function of \VA{}, right: the resulting train | |
125 | of signals from \eqref{eq:sim:sub_adc_counts}. Note, that the | |
126 | signal value used is just before the turn to the next value.} | |
127 | \label{fig:sim:va1_response} | |
128 | \end{figure} | |
129 | ||
130 | Since the ADC has a limited range of 10bits ($=10^2-1=1024-1$) all | |
131 | signals are truncated at 1023. | |
132 | ||
133 | For summable digits (\class{AliFMDSDigit}) $C_i=0$, but for | |
134 | fully simulated digits $c'_i$ (\class{AliFMDDigit}) it is given by | |
135 | the pedestal $p_i$ and noise $n_i$ of the strip | |
136 | ||
137 | \begin{equation} | |
138 | \label{eq:sim:pedestal_value} | |
139 | C_i = \text{gaus}(p_i,n_i)\quad[\N{}] | |
140 | \end{equation} | |
141 | that is, a Gaussian distributed number with $\mu=p_i$ (pedestal) and | |
142 | $\sigma=n_i$ (noise). | |
143 | ||
144 | \section*{Raw data} | |
145 | ||
146 | The raw data, whether from simulation or the experiment, is stored in | |
147 | the \ALTRO{}/\RCU{} data format. The \ALTRO{} has 10 bit (maximum | |
148 | count value of $\N{10^2-1}=\N{1023}$) ADC with up to 1024 consecutive | |
149 | samples of the input signal. The 128 input strip signals of \VA{} | |
150 | chips, are multiplexed into a single \ALTRO{} channel in such a way | |
151 | that each strip signal is sampled 1, 2, or 4 times\footnote{Currently, | |
152 | the default is to sample 2 times.}. | |
153 | ||
154 | The signal is then pedestal subtracted | |
155 | \begin{equation} | |
156 | \label{eq:sim:pedestal_subtraction} | |
157 | d_i = c_i - p_i + f_{ol} n_i\quad[\N{}] | |
158 | \end{equation} | |
159 | where $p_i$ and $n_i$ are the pedestal and noise value, evaluated | |
160 | on--line in special calibration runs, and $f_{ol}$ is a integer factor | |
161 | selected when configuring the detector\footnote{Typically $f_{ol}=4$.} | |
162 | ||
163 | After pedestal subtraction, which ensures that strips not hit by a | |
164 | particle has a 0 signals, an zero--suppression filter is applied by | |
165 | the \ALTRO{}. This filter throws away all 0s from the data and | |
166 | replaces them with markers that allows one to reconstruct the position | |
167 | of the remaining signals in the sample sequence. | |
168 | ||
169 | The signals from each \ALTRO{} input channel is then packed into | |
170 | blocks and shipped to the \RCU{} and eventually the data acquisition | |
171 | system. | |
172 | ||
173 | In simulations a similar filter is applied to the data to simulate the | |
174 | \ALTRO{} channels. The total signal from the a strip | |
175 | \eqref{eq:sim:adc_counts} is then given by | |
176 | \begin{equation} | |
177 | \label{eq:raw:sim_digits} | |
178 | c_i = \Delta_{i,mc} f_{i,mc} + \text{gaus}(p_i,n_i) - p_i - f_{ol} n_i \quad[\N{}] | |
179 | \end{equation} | |
180 | and similar for $c_{ij}$ \eqref{eq:sim:sub_adc_counts}. | |
181 | ||
182 | \section*{Reconstruction} | |
183 | ||
184 | When reconstructing of either simulated data or data from the | |
185 | experiment, the first thing is to read in the raw data stored in the | |
186 | \ALTRO{}/\RCU{} data format\footnote{There is an option to reconstruct | |
187 | from the simulated \class{AliFMDDigit} objects directly, in which | |
188 | case this step is skipped entirely.}. This is done by the | |
189 | \class{AliFMDRawReader} class. | |
190 | ||
191 | Depending on whether or not the data was zero--suppressed, the | |
192 | \class{AliFMDRawReader} code will do a pedestal subtraction, or add in | |
193 | the noise previously subtracted in the \ALTRO{} (or simulation there | |
194 | of) | |
195 | ||
196 | \begin{equation} | |
197 | \label{eq:reco:pedestal_subtraction} | |
198 | s'_i = c_i + C_i = c_i + \left\{ | |
199 | \begin{array}{cl} | |
200 | - p_i & \text{not zero--suppressed}\\ | |
201 | + f_{ol} n_i & \text{zero-suppressed} | |
202 | \end{array}\right.\quad[\N{}] | |
203 | \end{equation} | |
204 | where $f_{ol}$ is the noise factor applied by the \ALTRO{}\footnote{This | |
205 | factor is stored in the event header and read by the | |
206 | \class{AliFMDRawReader} --- thus ensuring consistency.}, and $p_i$ | |
207 | and $n_i$ are the pedestal and noise value of the strip in question. | |
208 | ||
209 | In the reconstruction it is possible (via a \class{AliFMDRecoParam} | |
210 | object) to specify a stronger noise suppression factor $f_{reco}$. If | |
211 | the signal $s'_i$ is smaller than the noise $n_i$ times the greater of | |
212 | the two noise suppression factors, it is explicitly set to 0 | |
213 | \begin{equation} | |
214 | \label{eq:reco:low_signal_cut} | |
215 | s_i = \left\{\begin{array}{cl} | |
216 | s'_i & s'_i > n_i \max{f_{ol},f_{reco}}\\ | |
217 | 0 & \text{otherwise} | |
218 | \end{array}\right.\quad[\N{}] | |
219 | \end{equation} | |
220 | ||
221 | We now have a signal $s_i$ which is akin to $f_{i,mc}\Delta_{i,mc}$ of | |
222 | \eqref{eq:raw:sim_digits}. We therefor calculate the energy loss in | |
223 | the $i^{\text{th}}$ strip using the factor | |
224 | \begin{equation} | |
225 | \label{eq:reco:conversion_factor} | |
226 | f_{i,reco} = \frac{\DeltaMip{}}{q_{mip} g_i} = f_{i,mc}^{-1} | |
227 | \quad[\N{}^{-1}\MeV{}] | |
228 | \end{equation} | |
229 | which is the inverse of \eqref{eq:sim:conversion_factor}, and the | |
230 | energy loss is then | |
231 | \begin{equation} | |
232 | \label{eq:reco:energy_loss} | |
233 | \Delta_{i,reco} = s_i f_{i,reco}\quad[\MeV{}] | |
234 | \end{equation} | |
235 | ||
236 | \section*{From energy loss to ADC counts and back} | |
237 | ||
238 | If we take \eqref{eq:sim:adc_counts} and \eqref{eq:reco:energy_loss} | |
239 | and assume | |
240 | \begin{itemize} | |
241 | \item that $s_i$ is not suppressed by \eqref{eq:reco:low_signal_cut} | |
242 | \item \eqref{eq:reco:pedestal_subtraction} removes the fluctuations | |
243 | put in \eqref{eq:sim:pedestal_value} | |
244 | \end{itemize} | |
245 | and put them together we get | |
246 | \begin{align} | |
247 | \label{eq:all:all} | |
248 | \Delta_{i,reco} &= s_i f_{i,reco}\nonumber\\ | |
249 | &= (c_i + C_i) \frac{\DeltaMip{}}{q_{mip} g_i}\nonumber\\ | |
250 | &= \Delta_{i,mc} f_{i,mc} f_{i,mc}^{-1}\nonumber\\ | |
251 | &= \Delta_{i,mc} | |
252 | \end{align} | |
253 | ||
254 | \section*{Some calculations} | |
255 | ||
256 | Assuming a typical energy loss of \unit[2.9]{MeV\,cm\textsuperscript{-1}} and | |
257 | applying \eqref{eq:sim:adc_counts} and | |
258 | \eqref{eq:sim:conversion_factor}, we get a signal value over pedestal of | |
259 | \begin{align} | |
260 | c_i &= \unit[2.9]{MeV\,cm^{-1}}\,T\, f_{i,mc}\nonumber\\ | |
261 | &= \MeV{0.0928}\frac{\Q{29.67}\ | |
262 | \unit[2.2]{N\,Q^{-1}}}{\MeV{0.124}}\nonumber\\ | |
263 | &= \MeV{0.0928}\ \unit[526.40]{N MeV\textsuperscript{-1}}\nonumber\\ | |
264 | &= \N{48.85} | |
265 | \end{align} | |
266 | ||
267 | \end{document} |