class AliTRDcluster: public AliCluster

 Default constructor

 Constructor for self constructing cluster. In this approach the information is inserted gradualy into the
 cluster and all dependencies are (re)calculated inside the cluster itself.

 A.Bercuci <A.Bercuci@gsi.de>

 Constructor

 Constructor from online tracklet

 Copy constructor

 Assignment operator

 Adds track index. Currently assumed that track is an array of
 size 9, and up to 3 track indexes are stored in fTracks[3].
 Indexes are sorted according to:
  1) index of max number of appearances is stored first
  2) if two or more indexes appear equal number of times, the lowest
     ones are stored first;

 Reset all member to the default value

 Returns the total charge from a not unfolded cluster

 Returns the error parameterization in the radial direction for TRD clusters as function of
 the calibrated time bin (tb) and optionally distance to anode wire (z). By default (no z information)
 the mean value over all cluster to wire distance is chosen.

 There are several contributions which are entering in the definition of the radial errors of the clusters.
 Although an analytic defition should be possible for the moment this is not yet available but instead a
 numerical parameterization is provided (see AliTRDclusterResolution::ProcessSigma() for the calibration
 method). The result is displayed in the figure below as a 2D plot and also as the projection on the drift axis.

 Here is a list of uncertainty components:
 - Time Response Function (TRF) - the major contribution. since TRF is also not symmetric (even if tail is
   cancelled) it also creates a systematic shift dependent on the charge distribution before and after the cluster.
 - longitudinal diffusion - increase the width of TRF and scales with square root of drift length
 - variation in the drift velocity within the drift cell

 Author
 A.Bercuci <A.Bercuci@gsi.de>

 Returns the error parameterization for TRD clusters as function of the drift length (here calibrated time bin tb)
 and optionally distance to anode wire (z) for the LUT r-phi cluster shape. By default (no z information) the largest
 value over all cluster to wire values is chosen.

 For the LUT method the dependence of s_y with x and d is obtained via a fit to the cluster to MC
 resolution. (see class AliTRDclusterResolution for more details). A normalization to the reference radial position
 x0 = 0.675 (tb=5 for ideal vd) is also applied (see GetSYprf()). The function is *NOT* calibration aware !
 The result is displayed in the figure below as a 2D plot and also as the projection on the drift axis. A comparison
 with the GAUS parameterization is also given

 For the GAUS method the dependence of s_y with x is *analytic* and it is expressed by the relation.


 The result is displayed in the figure below as function of the drift time and compared with the LUT parameterization.

 Author
 A.Bercuci <A.Bercuci@gsi.de>

 Parameterization of the r-phi resolution component due to cluster charge.
 The value is the offset from the nominal cluster resolution defined as the
 cluster resolution at average cluster charge (q0).



 The definition is *NOT* robust against gain fluctuations and thus two approaches are possible
 when residual miscalibration are available:
   - determine parameterization with a resolution matching those of the gain
   - define an analytic model which scales with the gain.

 For more details please see AliTRDclusterResolution::ProcessCharge()

 Author
 A.Bercuci <A.Bercuci@gsi.de>

 Parameterization of the cluster error in the r-phi direction due to charge sharing between
 adiacent pads. Should be identical to what is provided in the OCDB as PRF [TODO]

 The parameterization is obtained from fitting cluster resolution at phi=exb and |x-0.675|<0.225.
 For more details see AliTRDclusterResolution::ProcessCenter().

 Author
 A.Bercuci <A.Bercuci@gsi.de>

 Drift length correction [cm]. Due to variation of mean drift velocity along the drift region
 from nominal vd at xd->infinity. For drift velocity determination based on tracking information
 the correction should be negligible.

 TODO to be parametrized in term of drift velocity at infinite drift length
 A.Bercuci (Mar 28 2009)

 PRF correction for the LUT r-phi cluster shape.

 (Re)Calculate cluster position in the x direction in local chamber coordinates (with respect to the anode wire
 position) using all available information from tracking.
 Input parameters:
   t0 - calibration aware trigger delay [us]
   vd - drift velocity in the region of the cluster [cm/us]
   z  - distance to the anode wire [cm]. By default average over the drift cell width.
   q & xq - array of charges and cluster positions from previous clusters in the tracklet [a.u.]
 Output values :
   return x position of the cluster with respect to the
   anode wire using all tracking information

 The estimation of the radial position is based on calculating the drift time and the drift velocity at the point of
 estimation. The drift time can be estimated according to the expression:


 where t_0 is the delay of the trigger signal. t_cause is the causality delay between ionisation electrons hitting
 the anode and the registration of maximum signal by the electronics - it is due to the rising time of the TRF
 A second order correction here comes from the fact that the time spreading of charge at anode is the convolution of
 TRF with the diffusion and thus cross-talk between clusters before and after local clusters changes with drift length.
 t_TC is the residual charge from previous (in time) clusters due to residual tails after tail cancellation.
 This tends to push cluster forward and depends on the magnitude of their charge.

 The drift velocity varies with the drift length (and distance to anode wire) as described by cell structure simulation.
 Thus one, in principle, can calculate iteratively the drift length from the expression:


 In practice we use a numerical approach (AliTRDcluster::GetXcorr()) to correct for anisochronity obtained from MC
 comparison (see AliTRDclusterResolution::ProcessSigma()). Also the calibration of 0 approximation (no x dependence)
 for t_cause is obtained from MC comparisons and impossible to disentangle in real life from trigger delay.

 Author
 Alex Bercuci <A.Bercuci@gsi.de>

 Calculate, in tracking cooordinate system, the r-phi offset the cluster
 from the middle of the center pad. Three possible methods are implemented:
   - Center of Gravity (COG) see AliTRDcluster::GetDYcog()
   - Look-up Table (LUT) see AliTRDcluster::GetDYlut()
   - Gauss shape (GAUS) see AliTRDcluster::GetDYgauss()
 In addition for the case of LUT method position corrections are also
 applied (see AliTRDcluster::GetYcorr())

 Set variance of TRD cluster in the r-phi direction for each method.
 Parameters :
   - s2  - variance due to PRF width for the case of Gauss model. Replaced by parameterization in case of LUT.
   - dt  - transversal diffusion coeficient
   - exb - tg of lorentz angle
   - x   - drift length - with respect to the anode wire
   - z   - offset from the anode wire
   - tgp - local tangent of the track momentum azimuthal angle

 The ingredients from which the error is computed are:
   - PRF (charge sharing on adjacent pads)  - see AliTRDcluster::GetSYprf()
   - diffusion (dependence with drift length and [2nd order] distance to anode wire) - see AliTRDcluster::GetSYdrift()
   - charge of the cluster (complex dependence on gain and tail cancellation) - see AliTRDcluster::GetSYcharge()
   - lorentz angle (dependence on the drift length and [2nd order] distance to anode wire) - see AliTRDcluster::GetSX()
   - track angle (superposition of charges on the anode wire) - see AliTRDseedV1::Fit()
   - projection of radial(x) error on r-phi due to fixed value assumed in tracking for x - see AliTRDseedV1::Fit()

 The last 2 contributions to cluster error can be estimated only during tracking when the track angle
 is known (tgp). For this reason the errors (and optional position) of TRD clusters are recalculated during
 tracking and thus clusters attached to tracks might differ from bare clusters.

 Taking into account all contributions one can write the the TRD cluster error parameterization as:


 From this formula one can deduce a that the simplest calibration method for PRF and diffusion contributions is
 by measuring resolution at B=0T and phi=0. To disentangle further the two remaining contributions one has
 to represent s2 as a function of drift length.

 In the gaussian model the diffusion contribution can be expressed as:


 thus resulting the PRF contribution. For the case of the LUT model both contributions have to be determined from
 the fit (see AliTRDclusterResolution::ProcessCenter() for details).

 Author:
 A.Bercuci <A.Bercuci@gsi.de>

 Compare relevant information of this cluster with another one

 Print cluster information

 Store the pad corruption position code

 Code: 1 = left cluster
       2 = middle cluster;
       4 = right cluster

 Store the status of the corrupted pad

 Code: 2 = noisy
       4 = Bridged Left
       8 = Bridged Right
      32 = Not Connected

 Get COG position
 Used for clusters with more than 3 pads - where LUT not applicable

 Calculates the cluster position using the lookup table.
 Method provided by Bogdan Vulpescu.

 (Re)Calculate cluster position in the y direction in local chamber coordinates using all available information from tracking.

 Input parameters:
   s2 - sigma of gaussian parameterization (see bellow for the exact parameterization)
   W  - pad width
   xd - drift length (with respect to the anode wire) [cm]
   wt - omega*tau = tg(a_L)
 Output values :
   y1 and y2 - partial positions based on 2 pads clusters
   return y position of the cluster from all information

 Estimation of y coordinate is based on the gaussian approximation of the PRF. Thus one may
 calculate the y position knowing the signals q_i-1, q_i and q_i+1 in the 3 adiacent pads by:


 where W is the pad width, y_0 is the position of the center pad and s^2 is given by


 with s_0 being the PRF for 0 drift and track incidence phi equal to the lorentz angle a_L and the diffusion term
 being described by:


 with x being the drift length. The weights w_1 and w_2 are taken to be q_i-1^2 and q_i+1^2 respectively

 Authors
 Alex Bercuci <A.Bercuci@gsi.de>
 Theodor Rascanu <trascanu@stud.uni-frankfurt.de>

 Create the LUT