A fast and precise peak finder

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

2. 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

6. 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 ai are constant, i=1..3; • skiare 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

7. 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

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

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

11. 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

12. 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

13. Research Seminar, Fermi Lab November 2005

14. Research Seminar, Fermi Lab November 2005