Pulse shape reconstruction CMS ECAL Introduction the problem

Pulse shape reconstruction : CMS ECAL • Introduction – the problem – the tools – experimental pulse shape definition • Methods of reconstruction used – third degree polynomial fit – analytic function fit – experimental shape fit method – weights method • Results with LASER data (lab 11) – time measurements – number of samples : resolutions • fixed gain • free gain – linearity preliminary results • Conclusions 1

Introduction The problem : • Study of real data : LASER events (lab 11) • 10 to 30 crystals read with 25 samples per event , usually 2000 events analyzed per laser run. • We want to reconstruct the maximum amplitude and the time of maximum for each event. • The aim of the study is to be able to get fast algorithms working for the summer test beam in order to validate the monitoring with realistic and precise tools. 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 2

Introduction : tools ROOT converted files are read with standard lab 11 root code (pulse. C) The lab 11 standard code provides : (see nice WEB page : suncms 100. cern. ch): - automatic substraction of pedestals stored into root files (for a given gain ) - gain coefficient automatically taken into account (modes 0, 1, 2, 3 or free gains) - the “experimental pulse shape” (profile histo) for a given crystal is automatically read from a root file. • - light (PN’s ) value and TDC measurements are read from the event header (assuming that LASER light is stable for 2000 events (checked) we didn’t use the PN’s values for this study !! (charge ADC) ) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 3

tools : data and pedestals Data analyzed have been recorded in april-may 2002. Pedestals runs are always recorded before laser runs with all 4 gains and analyzed quasi online. Means and s are stored into root files. My. SQL database is then used when analyzing data to find the appropriate pedestal runs (+- 2 hours around laser run) , means and sigmas are then automatically read and used to correct ADC samples. s of pedestals (means around 600) gain 22 (33) : gain 6 (9) : gain 3 (3) : gain 1 (1) : s = 6 -10 s = 2 -4 s = 2 -3 s = 1 -3 are the following : 8 on average (170 Me. V) 3 on average 2. 5 on average 1. 5 on average Correlation matrix (error matrix) is stored and can also be used in all the methods. 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 4

Raw pulse shape 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 5

Experimental pulse shape : definition TDC : is measuring the time between the trigger (LASER light) and the first clock following(40 Mhz) this time is called the offset and is in sample unit between 0 and 1. TDC has a window of 50 picoseconds (we get 500 measurements in 1 sample ADC : 25 ns). The offset can be used to reconstruct a pulse shape and fill a profile histogram : the experimental pulse shape in X : number of the sample (0 -25) + offset in Y : ADC measurement This experimental pulse shape will be use as a reference pulse shape but is also used to fit any function with minuit in order to get an analytic function; it can also be used as we will see to reconstruct maximum amplitude. 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 6

pulse shape : offset definition Front edge is around 2. 5 samples = 75 ns 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 7

Offset definition : TDC = 1 - 0. 25 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 8

Offset definition : TDC = 1 - 0. 5 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 9

Offset definition : TDC = 1 - 0. 75 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 10

Test : analytic function with minuit (with experimental pulse shape) Fit 1 sample before max and 4 samples after max 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 11

the problem : reconstruction of events We get events with 25 ADC samples and we want to measure event by event the maximum amplitude and the time of arrival of this maximum amplitude. The algorithm has to be fast in order to process all events in a mimimum time. . Up to know a few methods have been developped in order to do the job , I will present here 3 of this methods which have been compared with the same events and inside the same program to avoid any biais … Before presenting the method I will introduce the 3 degree polynomial fit which is often used as starting point for the fits. 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 12

third degree polynomial fit – Adjust a third degree polynomial function on the 3 highest samples gives a first order estimation of maximum (and time) amplitude which is useful to get a starting point for other methods … – Resolution is around 0. 4 % BUT there is a systematic shift of the maximum amplitude reconstructed depending of the position of the maximum… 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 13

degree 3 polynom fit method used to get starting point 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 14

analytic function fit : . We have first to choose a function which reproduces the data. The 2 best candidates are : a) Electronic function (simulation) : F(ti) = Ai(1+ dt/ab)a e – dt/b with dt = ti-tmax b) Exponential function : F(ti) = Aie-(adt+x)/b with x = e(- adt-1) and dt = ti-tmax Function a) is the best candidate with lab 11 2002 data was function b) with 2001 data due to a shorter front edge (50 ns instead of 75 ns) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 15

analytic function fit : Method : reference -> Pierre Billoir [Note : LPC 84 -30] First step : Measure the general parameters a and b with a c 2 minimisation. We read 500 evts and iterate 4 times to get the 2 general parameters a and b (common to all events = shape) and the 2 particular parameters (ampli max and time of maximum) without using the tdc time. For electronic function : a = 1. 05 b = 2. 01 Second step : a and b are fixed and we use a fast algorithm using df derivatives to measure very quickly and for each event the amplitude maximum and the time of the maximum … Degree 3 polynomial fit is used to get starting values of the maximum amplitude and the time of maximum. The parameters a and b have been studied and fixed for all crytals and gains ! ( algorithm is 30 lines long and very fast …) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 16

electronic function fit : dependance vs offset is OK ! 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 17

weights method Reference : (Pascal Paganini CMS IN 2000/03) – The amplitude maximum of the signal is given by : • A = S i wi X i • i is the number of samples used • wi is the weight of the sample i • Xi is the ADC measurement of sample i But the weights Wi have to be calculated for a given shape and are dependant from an analytic function : 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 18

weights method : • Calculation of weights : • M. W – l. F – g. 1 = 0 (vector notation) – Where M is the electronic noise covariance matrix (symmetric matrix with autocorrelation properties included) dimension of M is n*n where n is the number of samples (to defined) – F is an analytic function calculated at time of sample i : W = (w 1 … wn ) F = (f 1. . . fn ) 1 = (1… 1 ) – l and g are found using the 2 constraints W. F =1 and W. 1= 0 – Matrix calculation is easy inside ROOT to get l and g l ( (M-1. F). F (M-1. F). 1 11 june 2002 (M-1. F). F (M-1. F). 1 ) ( g) = 1 0 ( ) CMS ECAL : Patrick Jarry ( Saclay) 19

weights method : To calculate weight , we have to : - find an analytic function F : we choose the following function F(ti) = (1+ dt/ab)a e – dt/b with dt = ti-tmax a and b are measured from experimental pulse shape with minuit calculation - know the time of arrival of the signal we are using the TDC measurement. - choose a number of samples to be used Then we calculate a set of weights per bin of 0. 1 ns (remark : bin of 1 ns are too big loss of resolution of 0. 5 %) : the amplitude maximum is given by A = Si w(i, ti) Xi 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 20

weights method : 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 21

experimental pulse shape method This method has been developped and studied by J. P Pansart and Jean Bourotte). The aim is to avoid an analytic function fit and to directly use the experimental pulse shape of the signal for each crystal and each gain. When a run is read , the experimental shape (profile) of the closest run (with right gain) for a given crystal is read from a root file and a c 2 minimization is done in order to extract the maximum amplitude and time of maximum. 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 22

Experimental shape method 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 23

Time measurements Time of the maximum of the signal has been measured for the different methods (except weights method) We can plot the time measurement without TDC (offset correction) , we must get a flat distribution knowing that the phase of the light relative to the clock is random. We can also correct the time measured with the offset (i. e : plot time+offset) to get an estimation of the resolutions in time with the different methods. 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 24

Time measurement : shape method no offset taken into account Systematic effect at the stop of TDC (front clock edge) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 25

Time measurement with offset correction 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 26

Time measurements 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 27

resolutions study Once we get the methods, we decided to study the resolutions measured with fixed gains (0, 1, 2, 3) and with free gains. remind : FPPA is able to run with a fixed gain , the signal is amplified with a factor which is gain dependant : mode 0 1 2 3 : : x x 1 3. 5 6. 2 22 (gain 1) (gain 3) (gain 9) (gain 33) In free gain (mode 99) , signal is amplified according to the amplitude of the sample … hardware modifications have been done last winter on FPPA in order to improve the performances…. 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 28

free gains : experimental pulse shape not corrected from gain coefficient 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 29

free gains : experimental pulse shape corrected from gain coefficient before hardware modifications 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 30

free gains : experimental pulse shape after hardware modifications 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 31

number of samples per gain (on average) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 32

data : lab 11 LASER In order to study the results with the differents methods we have recorded LASER data with light varying between minimum and maximum (0 -500 in Ge. V equivalence)… some scanning runs have been recorded with 50 subruns (different LASER light) in : - fixed gain (gain 1) - free gains We plot the maximum amplitude for each subrun and a gaussian fit is performed on the distribution in order to get the s (in Ge. V). The following plots will show the s of the gaussian fits measured for the different methods (1 point per subrun ) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 33

study : number of samples to be used The number of samples to use in our fits can be different for each method. Let’s call (I, J) : a choice where our fit : begin I samples before the maximum sample end J samples after the maximum sample It means that a choice (I, J) is using I+J+1 samples in the fit. First method analyzed : the weights method : Here is a plot showing the resolution ( s in Ge. V) obtained for some couples (2, 2) , (2, 10) , (2, 12) ( we had to generate a set of weights for each configuration of number of samples ) Remark : (1, 8) needs 10*10*250 = 25000 weights (1, 16) needs 18*18*250 = 81000 weights ! (stored into root files) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 34

weights method : start 2 samples before maximum 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 35

weights method : fixed gain start 1 sample before maximum 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 36

weights method : free gains start 1 sample before maximum weights method needs at least 12 samples 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 37

experimental pulse shape method : number of samples for fixed gain 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 38

Experimental shape method : number of samples for free gain Experimental shape method : Ok with 5 samples 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 39

Analytic function fit method : number of samples for fixed gain (mode 1) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 40

Analytic function fit method : number of samples with fre gains Analytic function method : ok with 6 samples better with more samples 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 41

Comparaisons of resolutions free gains 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 42

best resolutions for the 3 methods: free gains 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 43

best resolutions in fixed gain 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 44

best resolutions in fixed gain (s) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 45

free gains : resolutions study (crystal 19) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 46

crystal 19 : free gains resolution 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 47

Free gains : resolutions study (crystal 16) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 48

Free gains : resolutions study (crystal 16) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 49

Free gains : resolutions study (crystal 18) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 50

Free gains : resolutions study (crystal 18) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 51

Linearity study in fixed gain (crystal 19) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 52

Linearity study in free gains (crystal 19) 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 53

Linearity study in fixed gain (crystal 18) Weights method (green) in trouble at low E! 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 54

Linearity : free gains – fixed gain 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 55

Linearity free gains –fixed gain 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 56

Conclusions : We have studied 3 differents methods and we are able to reconstruct quickly the maximum amplitude of the ECAL signals in fixed and free gains. All the results presented here are preliminary more study is needed specially for the weights method. Many thanks to lab 11 team (Jean B. , Jean F. , Marc D. , Steve U. , Markus H. , Robert and others …) who did a great job in software and hardware to improve a lot the tools and the results and provided me a lot of nice data ! Thanks also for their disponibility and patience … 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 57

Conclusions : The study that I did show that the 3 very different methods have differents advantages and inconvenients : I) analytic function fit method : is fast need at least 6 samples is stable is dependant of the shape of the function 2 parameters for all crystals and gains II) weights method : is very fast when weights have been calculated need more samples : 12 -18 samples is not so stable… (0. 3% 0. 5%) is very strongly dependant of the analytic function used to calculate weights. ( is perfect in simulation with 4 samples ) III) experimental pulse shape method is idealistic , cannot do better needs only 3 -4 samples is stable but a bit heavy with a lot of crystals …. 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 58

Conclusions : Many systematics effects have been studied and some are not solve actually (FPPA effect), but a lot of work has to be done… Linearity is still under study. The methods can be improved , in particular the weights method. We have to be ready for H 4 summer beam tests : - study the H 4 LASER pulse shape and tune all the shape parameters (a, b) - study the PN’s pulse shape (much broader with sampling ADC) - study the linearity of the chain in H 4 - be ready with the electron pulse shape - study and take into account the noise if any ( we are ready to take into account correlated noise if any ) All this work has to be done before beam !!! 11 june 2002 CMS ECAL : Patrick Jarry ( Saclay) 59