# Presentation: 585287

## 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 823-827

-                         Research Seminar, Fermi Lab
-                                   November 2005

-                         Research Seminar, Fermi Lab
-                                   November 2005

-                         Research Seminar, Fermi Lab
-                                   November 2005