A fast and precise peak finder

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A fast and precise peak finder

• V. Buzuloiu
• (University POLITEHNICA Bucuresti)

Research Seminar, Fermi
Lab November
2005
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The problem in HEP

• When hit, a detector element produces a pulse of
  a known shape, with the peak and time position
  depending on the hit intensity and its instant.
• By sampling one gets a few samples over the
  duration of the pulse, usually 3-5 samples,
  equally spaced, but randomly shifted relatively
  to the peak.
• Given these samples the task is to find
• The maximum value (the peak or amplitude) of the
  pulse
• The peak position in time, relative to the
  sampling clock
• The computation of the peak value and time
  position must be done
• With high precision (e.g. 8 bit precision is
  achieved in 1)
• In real time, i.e. the computation time must not
  exceed the minimum time interval between two
  consecutive pulses.

Research Seminar, Fermi
Lab November
2005
3
The pulse (know shape)

• The mathematical description of the pulse-shaped
  signal is a real-valued function defined over a
  finite time slot
• It can be interpreted as a point in an infinite
  dimensional space

Research Seminar, Fermi
Lab November
2005
4
Pulse sampling

• When sampled, the signal is represented by a
  point (representative point) in a finite
  dimensional space (3D in this example).
• The representative point depends on the shift
  between the sampling clock pulses and the signal.
• If we want to extract a feature of the signal
  from a representative point we have to look for
  an invariant of the whole set of representative
  points.

Research Seminar, Fermi
Lab November
2005
5
Representative curve

• Let T be the sampling period. Then the
  representative curve is given by shifting the
  samples over an interval 0, T.
• Any invariant of the representative curve can be
  used to describe a feature of the signal, but
• The invariant must be easy to compute.
• The invariant must suit a family of signals,
  corresponding to excitations of different
  intensities.

Research Seminar, Fermi
Lab November
2005
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The Linear quasi-invariant (1)

• Assuming the representative curve of a signal is
  a plane curve, then all its points satisfy
• where v and ai are constant, i1..3
• ski are the samples in temporal order, k1.p
• Thus, v is an invariant of the curve, with an
  extremely simple form, suited for fast
  computation.
• If the signal shaping circuit is a linear one and
  (1) is true for a signal belonging to the family,
  then (1) is true for any signal in the family.
• For linear shaping circuits it is enough to
  analyze the representation of a standard pulse
  (normalized peak value).
• Unfortunately (1) does not hold for pulse
  signals, but

Research Seminar, Fermi
Lab November
2005
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The Linear quasi-invariant (2)

• We can try a plane which best fits the
  representative curve.

• The errors of approximating v are quite small a
  few percent.

Research Seminar, Fermi
Lab November
2005
8
The piece-wise linear quasi-invariant (1)

• If we the error constraints are stronger, the one
  plane approximation is not good enough.
• The representative curve is in a e- neighborhood
  of the plane if the relative error in v does not
  exceed e.
• We shall try to fit a few planes, each on a
  segment of the curve, so that for each segment
  the curve remains in an e-neighborhood of the
  corresponding plane.
• The piece-wise linear filter will thus be

Research Seminar, Fermi
Lab November
2005
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The piece-wise linear quasi-invariant (2)
Research Seminar, Fermi
Lab November
2005
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The piece-wise linear quasi-invariant (3)
Research Seminar, Fermi
Lab November
2005
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Conclusions

• A precise and efficient method for extracting a
  signal feature has been presented
• The method is applicable for extracting any
  feature of a signal with a known shape.
• There is no limitation to 2D signals. The same
  method can be used for determining peak amplitude
  and location on images, with a sub-pixel precision

Research Seminar, Fermi
Lab November
2005
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References

• 1 V. Buzuloiu, Real time recovery of the
  amplitude and shift of a pulse from its samples.
  CERN/LAA/RT92-015, April, 1992
• 2 V. Buzuloiu, A fast and precise peak finder
  for the pulses generated by the future HEP
  detectors. CHEP '92 , Annecy, France - pages
  823-827

Research Seminar, Fermi
Lab November
2005
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Research Seminar, Fermi
Lab November
2005
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Research Seminar, Fermi
Lab November
2005