Lightweight Algorithms for a CBM L1-Trigger

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Lightweight Algorithms for a CBM L1-Trigger

• Joachim Gläß
• Computer Engineering, University of Mannheim
• Contents
• STS Tracking
• Hough Transform
• MAPS Tracking
• Kalman Filter

September 9, 2004 Second FutureDAQ Workshop
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CBM STS Detector

• Silicon Tracking System
• 7 detector layers inside the dipole magnet gap
• max. 7 x, y, z-coordinates per track
• up to 1000 tracks
• mean interaction rate
• ca. 107 events/second
• online tracking for
• L1-trigger

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
3
STS TrackingHough-Transform of Parabolas
ltgt
rotated by q
2 (z sinq x cosq)
1

homogenous magnetic field 1 T
0.3
(z cosq x sinq)2
Pz
Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
4
STS TrackingHough-Transform of Parabolas
ltgt
rotated by q
2 (z sinq x cosq)
1

homogenous magnetic field 1 T
0.3
(z cosq x sinq)2
Pz
Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
5
STS TrackingHough-Transform of Parabolas
ltgt
rotated by q
2 (z sinq x cosq)
1

homogenous magnetic field 1 T
0.3
(z cosq x sinq)2
Pz
Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
6
STS Tracking3-D Hough-Histogram

• According to the three parameters of a track
• bending 1/Pz, angles q and g (Px/Pz, Py/Pz)
• g detector slice corresponds to one 2-D
  Hough-histogram
• g planes are overlapping (multiple scattering)

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
7
STS TrackingHW Implementation

• Decomposition of a 3D Hough transform in several
  2D Hough transforms
• 1. step
• sorting according to starting angle g with
  overlap (perpendicular to magnetic field)
• gt about straight line
• store hits according to (overlapping) g slices

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
8
STS TrackingHW Implementation

• Decomposition of a 3D Hough transform in several
  2D Hough transforms
• 1. step
• sorting according to starting angle g with
  overlap (perpendicular to magnetic field)
• gt about straight line
• store hits according to (overlapping) g slices
• 2. step
• 2D Hough histogram
• calculate subsequent or in parallel

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
9
STS TrackingHW Implementation

• possible implementation of 2D Hough histogram
  using FPGA and LUT
• input data -gt LUT -gt Hough curve
• systolic processing gt code curve with few bits
• 31 x 95 gt start 7 bits, 1 bit/row gt 37 bits
• logic cells for Hough histogram 25,000 30,000
• logic cells for peak search 5,000
• logic cells for LUT initialisation and
  access 5,000
• external memory 8 x (1M x 16)

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
10
STS TrackingHW Implementation

• Processing speed (rough estimations)
• 1 hit/cycle
• e.g. 10 Gb/s link with 64 bit/hit
• gt 150 x 106 hits/s
• 1 hit/cycle gt 150 MHz
• 1500 to 10000 hits/event gt 10µs to 100µs
• total number of processing units
• ca. 200 x 10 Gb/s links needed for STS
• gt ca. 200 units

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
11
STS TrackingSimulation Results

• Efficiency
• e found tracks/all tracks with P gt 1GeV/c
• g ghost tracks/processed tracks
• i identified tracks/processed tracks
• 31 x 95 x 383 e 95 , g 25 , i 45
• 63 x 191 x 255 e 93 , g 12 , i 65

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
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STS TrackingSimulation Results

• Precision of the reconstructed momentum
• 63 x 191 x 255

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
13
STS TrackingOutlook

• 2 layers MAPS, 5 layers STS
• 7 layers STS
• 63 x 191 x 255 e 93 , g 12 , i 65
• 5 layers STS (layer 3 to 7) tune peak shaping and
  peak finding
• 63 x 191 x 255 e 96 , g 32 , i 45

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
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STS TrackingOutlook

• Strip Detektors
• Hough curve -gt Hough plane
• AND perpenticular Hough planes -gt Hough curve
• stereo angle lt 90 ?

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
15
MAPS TrackingKalman Filter Track Following

• MAPS layer 1 and 2
• (monolithic active pixel sensors)
• high resolution lt 10 µm
• slow readout gt 10 µs
• pile up of ca. 100 events
• Kalman Filter track following
• track hits from L3 L5 as seed
• later Hough transform
• distance predicted real hit
• gt 95 within 500 µm

100 µm Si
100 µm Si
100 µm Si
Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
16
MAPS TrackingKalman Filter Track Following

• y-z plane (non-bending) gt straight line
• y m z c
• start with m0 y0/z0, c00
• predict position in previous layer yk mk-1 zk
  ck-1
• measure position (distance predicted real Dyk)
• update estimate with measurement
• mk (((mk-1zk-1)ck-1)-((mk-1zk)ck-1Dyk))/(zk
  -1-zk)
• ck (((mk-1zk)ck-1Dyk)-(mkzk))
• noise and error covariance are chosen to
  believe the latest measurement
• yk, mk, ck are function of mk-1, ck-1 and Dyk
• Dyk lt 500 µm gt needs few bits to code

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
17
MAPS TrackingKalman Filter Track Following

• x-z plane (magnetic field) gt parabola
• x a z2 b z c
• start with a0, b0 from hits in layer 3, 4, 5 (or
  Hough-Transform), c00
• predict position in previous layer xk ak-1 zk2
  bk-1 zk ck-1
• measure position (distance predicted real Dxk)
• update estimate with measurement
• ak f(ak-1,bk-1,ck-1,Dxk)
• bk f(ak-1,bk-1,ck-1,Dxk)
• ck f(ak-1,bk-1,ck-1,Dxk)
• xk, ak , bk , ck are function of ak-1 , bk-1 ,
  ck-1 and Dxk
• Dxk lt 500 µm gt needs few bits to code

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
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MAPS TrackingKalman Filter Track Following

• hits must be in layers 3, 4, 5
• no binning of data
• max distance 0.5 mm
• as function of PZ
• tracks with lower momentum
• are worse
• w/o pileup
• 98 of hits from same track
• with pileup
• no missing hits
• less hits from same track
• (ca. 10 )

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
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MAPS TrackingHardware Challence

• coefficients and parameters with 10 12 bit
  sufficient
• no double precision floating point needed
• old values -gt LUTs -gt adder -gt LUT -gt new value
• associative hit memory

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering
20
Summary

• Hough Transform
• global algorithm
• processing time number of hits
• possible implementation using FPGA and LUT
• efficiency ca. 95 of tracks found
• relatively high ghost rate
• able to handle strip detectors
• Kalman Filter
• MAPS pile up ca. 100 events
• w/o pile up ca. 98 of nearest hits from same
  track
• with pile up ca. 88 of nearest hits from same
  track
• ca. 12 of nearest hits from other events
• possible implementation using FPGA and LUT
• simple calculation
• associative hit memory

Joachim Gläß, Univ. Mannheim, Institute of
Computer Engineering