[B._Beckhoff,_et_al.]_Handbook_of_Practical_X-Ray_(b-ok.org).pdf

We must observe also that this kind of shaping is

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We must observe also that this kind of shaping is nonrealistic, due to the bilateral infinite width. Nevertheless, the results of the “optimum filtering” theory are very useful because they represent the limit to be approached by the practical filters. A discussion about practical filter is given in Section “Practical Signal Processing”. Practical Signal Processing A practical signal processor can only “approach” the optimum ENC obtained with the ideal optimum filters described in Section “Optimum Signal Process- ing” (p. 245). for any filter, characterized by the shape factors A 1 , A 2 , and A 3 , it can be seen from (4.25) and (4.30) that its optimum shaping time τ opt is related to the noise corner time constant τ c (which is the optimum shaping time for the ideal cusp-shaped filter) by τ opt τ c = A 1 A 3 . (4.34) By comparing (4.26) and (4.32) it is possible to see how worse is the ENC obtainable with the considered filter with respect to the one obtainable with an ideal filter. In particular, if only white noise is present, it turns out that ENC 2 w ( τ opt ) ENC 2 w opt = A 1 A 3 . (4.35) If also 1 /f noise is present, the worsening of the ENC with respect to the ideal case is less direct to see. It can be shown that the ENC obtainable with the considered filter satisfies the following inequality ENC 2 ( τ opt ) ENC 2 w opt (1 + K ) A 1 A 3 , (4.36) where the factor K , which is related to the amount of the 1 /f noise present, has been defined in (4.33). In the following the shape factors of some practical filters are given. Nowa- days, the most used filtering amplifier is the “semi-Gaussian” one based on a constellation of complex poles. Semi-Gaussian filters based on real poles and CR–RC filters can still be found in many laboratories. An important filter is the trapezoidal one, which can nowadays be also implemented as a time-variant parameter circuit. The triangular filter is important for didacti- cal reasons (it is easy to make mathematics and determine the shape factors with this filter as well as with CR–RC filter).
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X-Ray Detectors and XRF Detection Channels 249 It should be noted that nowadays digital signal processing (DSP) tech- niques allow practically the implementation of any weighting function, with any kind of shape constraint (e.g. finite width of the impulse response, flat top, etc.). See 4.2.11 Appendix 3 (pp. 259–262) for a short introduction to the digital filtering techniques in X-ray spectroscopy. 4.2.7 Shape Factors of some Filtering Amplifiers The ENC can be easily evaluated in the case of some practical signal proces- sors, once the “shape factors” are known. We consider a few typical cases and the values of the shape factors for each filter are given in Table 4.2. CR–RC Shaping The CR–RC filter is the simplest among the shaping amplifiers. Nowadays, it is very seldom used because other higher order filters giving a better ENC can be easily implemented. The output pulse of this shaper, fed by a unitary amplitude step-like pulse, is v uso ( t ) = t τ c exp t τ c , (4.37) where τ c = RC . The peak value of the output signal is obtained at
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