A fast and precise peak finder V. Buzuloiu (University POLITEHNICA Bucuresti) Research Seminar, Fermi Lab November 2005.

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Presentation transcript:

A fast and precise peak finder V. Buzuloiu (University POLITEHNICA Bucuresti) Research Seminar, Fermi Lab November 2005

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

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

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

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

The Linear quasi-invariant (1) Assuming the representative curve of a signal is a plane curve, then all it’s points satisfy:  where v and a i are constant, i=1..3;  s ki are the samples in temporal order, k=1….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

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

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

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

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

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

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 Research Seminar, Fermi Lab November 2005

Research Seminar, Fermi Lab November 2005

Research Seminar, Fermi Lab November 2005